Single-machine scheduling with learning effect and resource-dependent processing times

We consider resource allocation scheduling with learning effect in which the processing time of a job is a function of its position in a sequence and its resource allocation. The objective is to find the optimal sequence of jobs and the optimal resource allocation separately. We concentrate on two goals separately, namely, minimizing a cost function containing makespan, total completion time, total absolute differences in completion times and total resource cost; minimizing a cost function containing makespan, total waiting time, total absolute differences in waiting times and total resource cost. We analyse the problem with two different processing time functions. For each combination of these, we provide a polynomial time algorithm to find the optimal job sequence and resource allocation.     

Two-machine no-wait flowshop scheduling with learning effect and convex resource-dependent processing times

Two-machine no-wait flowshop scheduling problems in which the processing time of a job is a function of its position in the sequence and its resource allocation are considered in the study. The primary objective is to find the optimal sequence of jobs and the optimal resource allocation separately. Here we propose two separate models: minimizing a cost function of makespan, total completion time, total absolute differences in completion times and total resource cost; minimizing a cost function of makespan, total waiting time, total absolute differences in waiting times and total resource cost. Since each model is strongly NP-hard, we solve both models by breaking them down to two sub-problems, the optimal resource allocation problem for any job sequence and the optimal sequence problem with its optimal resource allocation. Specially, we transform the second sub-problem into the minimum of the bipartite graph optimal matching problem (NP-hard), and solve it by using the classic KM (Kuhn–Munkres) algorithm. The solutions of the two sub-problems demonstrate that the target problems remain polynomial solvable under the proposed model.     

Scheduling a batching machine with convex resource consumption functions

In various industries jobs undergo a batching, or burn in, process where different tasks are grouped into batches and processed simultaneously. The processing time of each batch is equal to the longest processing time among all jobs contained in the batch. All to date studies dealing with batching machines have considered fixed job processing times. However, in many real life applications job processing times are controllable through the allocation of a limited resource. The most common and realistic model assumes that there exists a non-linear and convex relationship between the amount of resource allocated to a job and its processing time. The scheduler?s task when dealing with controllable processing times is twofold. In addition to solving the sequencing problem, one must establish an optimal resource allocation policy. We combine these two widespread models on a single machine setting, showing that both the makespan and total completion time criteria can be solved in polynomial time. We then show that our proposed approach can be applied to general bi-criteria objective comprising of the makespan and the total completion time.     

Single-machine group scheduling with resource allocation and learning effect

This paper addresses single-machine scheduling problems under the consideration of learning effect and resource allocation in a group technology environment. In the proposed model of this paper the actual processing times of jobs depend on the job position, the group position, and the amount of resource allocated to them concurrently. Learning effect and two resource allocation functions are examined for minimizing the weighted sum of makespan and total resource cost, and the weighted sum of total completion time and total resource cost. We show that the problems for minimizing the weighted sum of makespan and total resource cost remain polynomially solvable. We also prove that the problems for minimizing the weighted sum of total completion time and total resource cost have polynomial solutions under certain conditions.     

Note on a unified approach to the single-machine scheduling problem with a deterioration effect and convex resource allocation

In many resource allocation problems in physical or economic systems, a linear resource consumption function is commonly considered, and job processing times are assumed to be fixed parameters. However, the former assumption fails to reflect the law of diminishing returns, and the latter may be controlled by changing the allocation of resources to jobs. Motivated by these observations, we provide a unified model for solving single-machine scheduling problems in which each job's processing time is a function of its starting time and convex resource allocation. The objective is to find the optimal sequence of jobs subject to a limited resource consumption. We first show how this unified model can be useful in solving scheduling problems under due date assignment considerations. We analyze the problem with four different due date assignment methods, and our objective function includes costs for earliness, tardiness and due date assignments. We also consider scheduling problems without involving due date assignment decisions. The objective function is to minimize the makespan, total completion time, total absolute variation in completion times, and total absolute variation in waiting times. We show that several existing well-known problems can be reduced to a special case of our unified model and solved in O(nlogn) time.     

到达时间依赖于资源分配的单机排序问题*

研究了具有线性退化及学习效应作用下的单机排序问题,对于工件的到达时间是其资源消耗量的正的严格单调递减函数时,考虑了总资源消耗量限定情形下最大完工时间极小化问题,给出了相应的最优算法;也考虑了满足工件最大完工时间限制的条件下极小化资源消耗的总量问题,提出最优资源分配方案。     

Single-machine group scheduling with both learning effects and deteriorating jobs

In the paper two resource constrained single-machine group scheduling problems with both learning effects and deteriorating jobs are considered. By learning effects, deteriorating jobs and group technology assumption, we mean that the processing time of a job is defined by the function of its starting time and position in the group, and the group setup times of a group is a positive strictly decreasing continuous function of the amount of consumed resource. We present polynomial solutions for the makespan minimization problem under the constraint that the total resource consumption does not exceed a given limit, and the total resource consumption minimization problem under the constraint that the makespan does not exceed a given limit, respectively.     

A bicriterion approach to time/cost trade-offs in scheduling with convex resource-dependent job processing times and release dates

This paper presents a bicriterion analysis of time/cost trade-offs for the single-machine scheduling problem where both job processing times and release dates are controllable by the allocation of a continuously nonrenewable resource. Using the bicriterion approach, we distinguish between our sequencing criterion, namely the makespan, and the cost criterion, the total resource consumed, in order to construct an efficient time/cost frontier. Although the computational complexity of the problem of constructing this frontier remains an open question, we show that the optimal job sequence is independent of the total resource being used; thereby we were able to reduce the problem to a sequencing one. We suggest an exact dynamic programming algorithm for solving small to medium sizes of the problem, while for large-scale problems we present some heuristic algorithms that turned out to be very efficient. Five different special cases that are solvable by using polynomial time algorithms are also presented.     

Single-machine batch delivery scheduling with an assignable common due date and controllable processing times

We consider single-machine batch delivery scheduling with an assignable common due date and controllable processing times, which vary as a convex function of the amounts of a continuously divisible common resource allocated to individual jobs. Finished jobs are delivered in batches and there is no capacity limit on each delivery batch. We first provide an O(n⁵) dynamic programming algorithm to find the optimal job sequence, the partition of the job sequence into batches, the assigned common due date, and the resource allocation that minimize a cost function based on earliness, tardiness, job holding, due date assignment, batch delivery, and resource consumption. We show that a special case of the problem can be solved by a lower-order polynomial algorithm. We then study the problem of finding the optimal solution to minimize the total cost of earliness, tardiness, job holding, and due date assignment, subject to limited resource availability, and develop an O(nlog n) algorithm to solve it.     

Two-machine flow-shop scheduling with rejection

We study a scheduling problem with rejection on a set of two machines in a flow-shop scheduling system. We evaluate the quality of a solution by two criteria: the first is the makespan and the second is the total rejection cost. We show that the problem of minimizing the makespan plus total rejection cost is NP-hard and for its solution we provide two different approximation algorithms, a pseudo-polynomial time optimization algorithm and a fully polynomial time approximation scheme (FPTAS). We also study the problem of finding the entire set of Pareto-optimal points (this problem is NP-hard due to the NP-hardness of the same problem variation on a single machine [20]). We show that this problem can be solved in pseudo-polynomial time. Moreover, we show how we can provide an FPTAS that, given that there exists a Pareto optimal schedule with a total rejection cost of at most R and a makespan of at most K, finds a solution with a total rejection cost of at most (1+?)R and a makespan value of at most (1+?)K. This is done by defining a set of auxiliary problems and providing an FPTAS algorithm to each one of them.     

Scheduling jobs with simultaneous considerations of controllable processing times and learning effect

This paper considers a scheduling problem with general job-dependent learning curves and controllable processing times on a single machine. The objective is to determine the optimal compressions of the processing times and the optimal sequence of jobs so as to minimize some total cost functions, which consist of regular and non-regular functions and the processing time compressions. It shows that the problem can be solved by an assignment problem and thus can be solved in polynomial time. Some extensions of the problem are also given.

     

Pareto optimization of rescheduling with release dates to minimize makespan and total sequence disruption

We consider the rescheduling on a single machine with release dates to minimize the makespan and total sequence disruption simultaneously. In the literature, a polynomial-time algorithm was presented for minimizing the makespan under a limit on the total sequence disruption. But the algorithm is not strongly polynomial. We present a strongly polynomial-time algorithm for finding all Pareto optimal points of the Pareto optimization problem. Consequently, the rescheduling to minimize the makespan under a limit on the total sequence disruption can be solved in a strongly polynomial time.     

Resource-dependent scheduling with deteriorating jobs and learning effects on unrelated parallel machine

The focus of this paper is to analyze unrelated parallel-machine resource allocation scheduling problem with learning effect and deteriorating jobs. The goal is to find the optimal sequence of jobs and the optimal resource allocation separately for minimizing the cost function including the total load, the total completion time, the total absolute deviation of completion time and the total resource cost. We show that the problem is polynomial time solvable if the number of machines is a given constant.     

Utility-driven solution for optimal resource allocation in computational grid

Optimal resource allocation is a complex undertaking due to large-scale heterogeneity present in computational grid. Traditionally, the decision based on certain cost functions has been used in allocating grid resource as a standard method that does not take resource access cost into consideration. In this paper, the utility function is presented as a promising method for grid resource allocation. To tackle the issue of heterogeneous demand, the user's preference is represented by utility function, which is driven by a user-centric scheme rather than system-centric parameters adopted by cost functions. The goal of each grid user is to maximize its own utility under different constraints. In order to allocate a common resource to multiple bidding users, the optimal solution is achieved by searching the equilibrium point of resource price such that the total demand for a resource exactly equals the total amount available to generate a set of optimal user bids. The experiments run on a Java-based discrete-event grid simulation toolkit called GridSim are made to study characteristics of the utility-driven resource allocation strategy under different constraints. Results show that utility optimization under budget constraint outperforms deadline constraint in terms of time spent, whereas deadline constraint outperforms budget constraint in terms of cost spent. The conclusion indicates that the utility-driven method is a very potential candidate for the optimal resource allocation in computational grid.     

Single-machine scheduling with deteriorating jobs

In this article, we study a single-machine scheduling problem in which the processing time of a job is a nonlinear function of its basic processing time and starting time. The objectives are to minimise the makespan, the sum of weighted completion times and the sum of the kth powers of completion times. We show that the makespan minimisation problem can be solved in polynomial time. However, the total completion time and the sum of the kth powers of completion times minimisation problems can be solved in polynomial time in some cases. Besides, some useful properties are also provided for the sum of weighted completion times problem under certain conditions.     

一类资源约束的单机成组调度问题

本文讨论具有连续资源的单机成组调度问题.在这一模型中,工件组的安装时间是所消耗资源的非负严格减少连续函数,工件的加工时间是开工时间的严格增加函数.考虑两个问题,第1个问题是在满足资源消耗总量限制条件下,极小化最大完工时间.第2个问题的目标函数是在满足最大完工时间限制条件下,极小化资源消耗总量.分别对两个问题讨论了最优调度的某些特征,分别给出了求解最优资源分配的方法,并通过数值例子进行说明.     

A two-stage flow shop scheduling problem on a batching machine and a discrete machine with blocking and shared setup times

Motivated by applications in iron and steel industry, we consider a two-stage flow shop scheduling problem where the first machine is a batching machine subject to the blocking constraint and the second machine is a discrete machine with shared setup times. We show that the problem is strongly NP-hard when the objective is to minimize the makespan. When solved with a heuristic priority rule, the worst case ratio with the minimum makespan is 2. For a more general objective, the minimization of a linear combination of the makespan and the total blocking time, a quadratic mixed integer program is presented first. Then we pinpoint two cases with polynomial time algorithms: the case without blocking constraint and the case with a given job sequence. Also for the general objective, we analyze an approximation algorithm. Finally, we evaluate the algorithms, giving experimental results on randomly generated test problems.     

Single‐machine common flow allowance scheduling with aging effect,resource allocation,and a rate‐modifying activity

In this paper, we consider scheduling of deteriorating jobs on a single machine with slack (SLK) due date assignment, resource allocation, and a rate‐modifying activity. The rate‐modifying activity can change jobs’ processing rates such that the actual processing time of a job depends on whether the job is processed before or after the rate‐modifying activity. In addition, the actual processing time of a job also depends on its position in a processing sequence (i.e., the aging effect) and the amount of resource allocated to it. The objective is to determine the optimal sequence, optimal common flow allowance, optimal resource allocation, and optimal location of the rate‐modifying activity to minimize a total penalty function comprising the earliness, tardiness, common flow allowance, and resource allocation costs. We consider two variants of the problem associated with two different processing time functions and provide a polynomial‐time algorithm to solve each variant.     

Joint object placement and node dimensioning for Internet content distribution

This paper studies a resource allocation problem in a graph, concerning the joint optimization of capacity allocation decisions and object placement decisions, given a single capacity constraint. This problem has applications in Internet content distribution and other domains. The solution to the problem comes through a multi-commodity generalization of the single commodity k-median problem. A two-step algorithm is developed that is capable of solving the multi-commodity case optimally in polynomial time for the case of tree graphs, and approximately (within a constant factor of the optimal) in polynomial time for the case of general graphs.     

具有恶化效应和学习效应的单机成组调度问题

讨论了一类具有恶化效应和学习效应的单机成组调度问题, 其中工件的加工时间为开工时间和组内工序的函数. 通过对问题性质的分析以及多项式时间算法的描述, 得出如下结论: 在单机成组调度问题中, 即便工件的加工时间同时受恶化效应和学习效应的制约, 极小化完工时间问题以及极小化总资源消耗的问题仍是多项式时间可解的.