A note on worst-case performance of heuristics for maintenance scheduling problems

We study a machine scheduling model in which job scheduling and machine maintenance activities have to be considered simultaneously. We develop the worst-case bounds for some heuristic algorithms, including a sharper worst-case bound of the SPT schedule than the results in the literature, and another bound of the EDD schedule.     

Single machine scheduling with exponential learning functions

In this paper we consider the single machine scheduling problem with exponential learning functions. By the exponential learning functions, we mean that the actual job processing time is a function of the total normal processing times of the jobs already processed. We prove that the shortest processing time (SPT) rule is optimal for the total lateness minimization problem. For the following three objective functions, the total weighted completion time, the discounted total weighted completion time, the maximum lateness, we present heuristic algorithms according to the corresponding problems without exponential learning functions. We also analyse the worst-case bound of our heuristic algorithms. It also shows that the problems of minimizing the total tardiness and discounted total weighted completion time are polynomially solvable under some agreeable conditions on the problem parameters.     

Single-machine total completion time scheduling with a time-dependent deterioration

In this paper, we consider the single-machine scheduling problems with a time-dependent deterioration. By the time-dependent deterioration, we mean that the processing time of a job is defined by an increasing function of total normal processing time of jobs in front of it in the sequence. The objective is to minimize the total completion time. We develop a mixed integer programming formulation for the problem. The complexity status of this problem remains open. Hence, we use the smallest normal processing time (SPT) first rule as a heuristic algorithm for the general cases and analyze its worst-case error bound. Two heuristic algorithms utilize the V-shaped property are also proposed to solve the problem. Computational results are presented to evaluate the performance of the proposed algorithms.     

Single-machine scheduling problems with time and position dependent processing times

We consider single-machine scheduling problems with time and position dependent job processing times. In many industrial settings, the processing time of a job changes due to either job deterioration over time or machine/worker’s learning through experiences. In the models we study, each job has its normal processing time. However, a job’s actual processing time depends on when its processing starts and how many jobs have completed before its start. We prove that the classical SPT (Shortest Processing Time) rule remains optimal when we minimize the makespan or the total completion time. For problems of minimizing the total weighted completion time, the maximum lateness, and the discounted total weighted completion time, we present heuristic sequencing rules and analyze the worst-case bounds for performance ratios. We also show that these heuristic rules can be optimal under some agreeable conditions between the normal processing times and job due dates or weights.     

Learning heuristics for basic block instruction scheduling

Instruction scheduling is an important step for improving the performance of object code produced by a compiler. A fundamental problem that arises in instruction scheduling is to find a minimum length schedule for a basic block—a straight-line sequence of code with a single entry point and a single exit point—subject to precedence, latency, and resource constraints. Solving the problem exactly is known to be difficult, and most compilers use a greedy list scheduling algorithm coupled with a heuristic. The heuristic is usually hand-crafted, a potentially time-consuming process. In contrast, we present a study on automatically learning good heuristics using techniques from machine learning. In our study, a recently proposed optimal basic block scheduler was used to generate the machine learning training data. A decision tree learning algorithm was then used to induce a simple heuristic from the training data. The automatically constructed decision tree heuristic was compared against a popular critical-path heuristic on the SPEC 2000 benchmarks. On this benchmark suite, the decision tree heuristic reduced the number of basic blocks that were not optimally scheduled by up to 55% compared to the critical-path heuristic, and gave improved performance guarantees in terms of the worst-case factor from optimality.     

Scheduling independent jobs with stochastic processing times and a common due date on parallel and identical machines

We propose a heuristic procedure that constructs a schedule forN jobs with stochastic processing times and a common due date onM parallel, identical machines. The criterion is the minimization of the total expected incompletion cost. A worst-case analysis for the ratio of the heuristic and optimal solutions is presented and a bound on the ratio is derived. The experimental results presented indicate that the heuristic procedure generates almost optimal solutions.     

Minimum and Worst-Case Performance Ratios of Rollout Algorithms

Rollout algorithms are heuristic algorithms that can be applied to solve deterministic and stochastic dynamic programming problems. The basic idea is to use the cost obtained by applying a well known heuristic, called the base policy, to approximate the value of the optimal cost-to-go. We develop a theoretical approach to prove, for the 0-1 knapsack problem, that the minimum performance ratio of the rollout algorithms tends to be significantly greater when the performance ratio of the corresponding base policy is poor and that the worst-case performance ratio is significantly better than the one of the corresponding base policies.     

Single-job lot streaming in m − 1 two-stage hybrid flowshops

This paper studies the single-job lot streaming problem in a two-stage hybrid flowshop that has m identical machines at the first stage and one machine at the second stage, to minimise the makespan. A setup time is considered before processing each sublot on a machine. For the problem with the number of sublots given, we prove that it is optimal to use a rotation method for allocating and sequencing the sublots on the machines. With such allocation and sequencing, the sublot sizes are then optimised using linear programming. We then consider the problem with equal sublot sizes and develop an efficient solution to determining the optimal number of sublots. Finally optimal and heuristic solution methods for the general problem are proposed and the worst-case performance of the equal-sublot solution is analysed. Computational results are also reported demonstrating the close-to-optimal performances of the heuristic methods in different problem settings.     

A note on the minimum bounded edge-partition of a tree

Minimum bounded edge-partition divides the edge set of a tree into the minimum number of disjoint connected components given a maximum weight for any component. It is an adaptation of the uniform edge-partition of a tree. An optimization algorithm is developed for this NP-hard problem, based on repeated bin packing of inter-related instances. The algorithm has linear running time for the class of ‘balanced trees’ common for the stochastic programming application which motivated investigation of this problem.Fast 2-approximation algorithms are formed for general instances by replacing the optimal bin packing with almost any bin packing heuristic. The asymptotic worst-case ratio of these approximation algorithms is never better than the absolute worst-case ratio of the bin packing heuristic used.     

双机成比例无等待流水线重调度干扰管理研究

针对由异速机构成的双机成比例无等待流水线的加工特点,研究了机器扰动工况下的生产重调度问题,提出了兼顾初始调度目标(最小化制造期)和扰动修复目标(最小化工件滞后时间和)的干扰管理方法。在最短加工时间优先(SPT)排序规则的最优解特性分析基础上,证明了右移初始加工时间表是事后干扰管理的最优调度方案,建立了基于SPT规则的事前干扰管理模型,设计了基于理想点趋近的多目标处理策略,提出了离散量子微粒群优化与局部搜索机制相结合的启发式模型求解算法。算例实验结果表明,本文提出的干扰管理模型和算法是有效的。     

Approximation algorithms for two-machine flow shop scheduling with batch setup times

In many practical situations, batching of similar jobs to avoid setups is performed while constructing a schedule. This paper addresses the problem of non-preemptively scheduling independent jobs in a two-machine flow shop with the objective of minimizing the makespan. Jobs are grouped into batches. A sequence independent batch setup time on each machine is required before the first job is processed, and when a machine switches from processing a job in some batch to a job of another batch. Besides its practical interest, this problem is a direct generalization of the classical two-machine flow shop problem with no grouping of jobs, which can be solved optimally by Johnson's well-known algorithm. The problem under investigation is known to be NP-hard. We propose two O(n logn) time heuristic algorithms. The first heuristic, which creates a schedule with minimum total setup time by forcing all jobs in the same batch to be sequenced in adjacent positions, has a worst-case performance ratio of 3/2. By allowing each batch to be split into at most two sub-batches, a second heuristic is developed which has an improved worst-case performance ratio of 4/3. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Joint Replenishment in Multi-Item Inventory Systems

A multi-item inventory system is considered which has the property that, for each single item, a reorder policy using the E.O.Q. formula would be appropriate. Holding costs are linear, and fixed ordering costs are assumed to be composed of a major set-up cost reflecting the pure fact of placing an order, and a sum of minor set-up costs corresponding to the items included in the order. If it is desirable to form a certain number of groups of items where all items of one group share the same order cycle, it is shown that there is always an optimal grouping in which items are arranged in increasing order of their ratio of yearly holding costs and minor set-up costs.A heuristic for forming the groups is given which turns out to be an optimal algorithm for the case that there are no major set-up costs. After an initial sorting of ratios, the worst-case complexity of this procedure is linear in the number of items.     

Worst-case performance of Wong's Steiner tree heuristic

A study of the worst-case performance of Wong's heuristic for the Steiner problem in directed networks (SPDN) is presented in this paper.SPDN is a classic combinatorial optimization problem having the status of a very difficult problem (NP-hard problem) and it is known as an optimization model for a broad class of problems in networks. Several exact and heuristic approaches have been designed for SPDN in the last twenty five years.Some papers analyze theoretical and experimental behavior of heuristics for SPDN, specially for undirected networks, but none of these has studied the worst-case performance of Wong's heuristic. In this paper, we find a lower bound for that performance and show that this bound is consistent with comparable results in the literature on SPDN and its undirected version.     

Some results of the worst-case analysis for flow shop scheduling with a learning effect

This article considers flow shop scheduling problems with a learning effect. By the learning effect, we mean that the processing time of a job is defined by a function of its position in a processing permutation. The objective is to minimize the total weighted completion time. Some heuristic algorithms by using the optimal permutations for the corresponding single machine scheduling problems are presented, and the worst-case bound of these heuristics are also analyzed.     

Analysis of a linear programming heuristic for scheduling unrelated parallel machines

Each of n jobs is to be processed without interruption on one of m unrelated parallel machines. The objectives is to minimize the maximum completion time. A heuristic method is presented, the first stage of which uses linear programming to form a partial schedule leaving at most m?1 jobs unscheduled: the second stage schedules these m?1 jobs using an enumerative method. For m≥3, it is shown that the heuristic has a (best possible) worst-case performance ratio of 2 and has a computational requirement which is polynomial in n although it is exponential in m. For m = 2, it is shown that the heuristic has a (best possible) worst-case performance ratio of $\text{1 +}\text{5}\text{)}\text{2}$ and requires linear time. A modified version of the heuristic is presented for m = 2 which is shown to have a (best possible) worst-case performance ratio of $\text{3}\text{2}$ while still requiring linear time.     

Worst-case analysis for flow shop scheduling problems with an exponential learning effect

A real industrial production phenomenon, referred to as learning effects, has drawn increasing attention. However, most research on this issue considers only single machine problems. Motivated by this limitation, this paper considers flow shop scheduling problems with an exponential learning effect. By the exponential learning effect, we mean that the processing time of a job is defined by an exponent function of its position in a processing permutation. The objective is to minimize one of the four regular performance criteria, namely, the total completion time, the total weighted completion time, the discounted total weighted completion time, and the sum of the quadratic job completion times. We present heuristic algorithms by using the optimal permutations for the corresponding single-machine scheduling problems. We also analyse the worst-case bound of our heuristic algorithms.     

A note on makespan minimization in two-stage flexible flow shops with uniform machines

We consider the two-stage flexible flow shop makespan minimization problem with uniform parallel machines. Soewandi and Elmaghraby [Soewandi, H., Elmaghraby, S., 2003. Sequencing on two-stage hybrid flowshops with uniform machines to minimize makespan. IIE Transaction 35, 467–477] developed a heuristic (S–E) and derived a machine speed-dependent worst-case ratio bound for it. We point out that this bound works well when the uniform machines have approximately equal speeds but is not indicative of the performance of the S–E heuristic when the machine speeds are in a wide range. Motivated by this observation, we propose an alternative tight machine-speed dependent worst-case bound for the S–E heuristic that works well when the machine speeds vary significantly. We then combine the two speed-dependent ratio bounds into a speed-independent bound. Our findings facilitate the narrowing of the gap between experimental performance and worst-case bound for the S–E heuristic.     

加工时间和尺寸成正比单机排序和工件运输的最小化最大完工时间问题

在单机排序和工件运输的最小化最大完工时间问题中,工件首先在一台机器上加工,然后被一辆有容量限制的汽车运送到一个顾客．当工件的加工时间和尺寸无关时, Chang和Lee已经证明该问题是强NP困难的．他们也给出了一个启发式算法,它的最差执行比为5／3,并且这个界是紧的．本文考虑工件的加工时间和尺寸成正比的情形,证明了Chang和Lee的算法有更好的最差执行比53／35,并提供了一个新的启发式算法,它的最差执行比是3／2,并且这个界是最好的．     

A linear programming approach for linear programs with probabilistic constraints

We study a class of mixed-integer programs for solving linear programs with joint probabilistic constraints from random right-hand side vectors with finite distributions. We present greedy and dual heuristic algorithms that construct and solve a sequence of linear programs. We provide optimality gaps for our heuristic solutions via the linear programming relaxation of the extended mixed-integer formulation of Luedtke et al. (2010) [13] as well as via lower bounds produced by their cutting plane method. While we demonstrate through an extensive computational study the effectiveness and scalability of our heuristics, we also prove that the theoretical worst-case solution quality for these algorithms is arbitrarily far from optimal. Our computational study compares our heuristics against both the extended mixed-integer programming formulation and the cutting plane method of Luedtke et al. (2010) [13]. Our heuristics efficiently and consistently produce solutions with small optimality gaps, while for larger instances the extended formulation becomes intractable and the optimality gaps from the cutting plane method increase to over 5%.     

An improved heuristic for two-machine flow shop scheduling with an availability constraint and nonresumable jobs

In this paper, we deal with the two-machine flow shop scheduling problem having an unavailability interval on the first machine, and nonresumable jobs. We first present an enhancement procedure that, once applied to any arbitrary solution, produces a schedule that is at most equal 2 times the optimal makespan. We then develop an improved heuristic, with a relative worst-case error of 3/2.