Single-machine due window assignment and scheduling with a common flow allowance and controllable job processing time

In this paper, we consider single-machine due window assignment and scheduling with a common flow allowance and controllable job processing times, subject to unlimited or limited resource availability. Due window assignment with a common flow allowance means that each job has a job-dependent due window, the starting time and completion time of which are equal to its actual processing time plus the job-independent parameters q₁ and q₂, respectively, which are common to all the jobs. The processing time of each job is either a linear or a convex function of the amount of a common continuously divisible resource allocated to the job. We study five versions of the problem that differ in terms of the objective function and processing time function being used. We provide structural properties of the optimal schedules and polynomial-time solution algorithms for the considered problems.     

Single machine scheduling problems with controllable processing times and total absolute differences penalties

In this paper, we consider single machine scheduling problem in which job processing times are controllable variables with linear costs. We concentrate on two goals separately, namely, minimizing a cost function containing total completion time, total absolute differences in completion times and total compression cost; minimizing a cost function containing total waiting time, total absolute differences in waiting times and total compression cost. The problem is modelled as an assignment problem, and thus can be solved with the well-known algorithms. For the case where all the jobs have a common difference between normal and crash processing time and an equal unit compression penalty, we present an O(n log n) algorithm to obtain the optimal solution.     

Optimal assignment of NOP due-dates and sequencing in a single machine shop

We consider the problem of optimal assignment of NOP due-dates ton jobs and sequencing them on a single machine to minimize a penalty function depending on the values of assigned constant waiting allowance and maximum job tardiness. It is shown that the earliest due date (EDD) order is an optimal sequence. For finding optimal constant waiting allowance, we reduce the problem to a multiple objective piecewise linear programming with single variable. An efficient algorithm for optimal solution of the problem is given.     

Single machine common flow allowance scheduling with deteriorating jobs and a rate-modifying activity

We study a single machine slack due date assignment (usually referred to as SLK) scheduling problem with deteriorating jobs and a rate-modifying activity. The deterioration effect manifest such that the job processing time is a function of its starting time in a sequence. The rate-modifying activity is an activity that changes the processing rate of machine, i.e., the machine performs a rate-modifying activity. Hence the actual processing time of a job is a variable, which depends not only on its starting time in a sequence but also on whether it is scheduled before or after a rate-modifying activity. The goal is to schedule the rate-modifying activity, the optimal common flow allowance and the sequence of jobs to minimize the total earliness, the total tardiness and the common flow allowance cost. We show that the problem remains polynomially solvable under the proposed model.     

Single-machine scheduling with common due-window assignment for deteriorating jobs

In this paper, we consider a single-machine common due-window assignment scheduling problem with deteriorating jobs. Jobs’ processing times are defined by function of their starting times and job-dependent deterioration rates that are related to jobs and are not all equal. The objective is to determine an optimal combination of sequence and common due-window location so as to minimize the weighted sum of earliness, tardiness and due-window location penalties. We propose an O(n² log n) time algorithm to solve the problem and discuss several instances to illustrate it.     

Optimal due-date assignment problem with learning effect and resource-dependent processing times

In this paper, we consider a single-machine earliness-tardiness scheduling problem with due-date assignment, in which the processing time of a job is a function of its position in a sequence and its resource allocation. The due date assignment methods studied include the common due date, and the slack due date, which reflects equal waiting time allowance for the jobs. For each combination of due date assignment method and processing time function, we provide a polynomial-time algorithm to find the optimal job sequence, due date values, and resource allocations that minimize an integrated objective function, which includes earliness, tardiness, due date assignment, and total resource consumption costs.     

The two-machine no-wait general and proportionate open shop makespan problem

We consider the two-machine no-wait open shop minimum makespan problem in which the determination of an optimal solution requires an optimal pairing of the jobs followed by the optimal sequencing of the job pairs. We show that the required enumeration can be curtailed by reducing the pair sequencing problem for a given pair set to a traveling salesman problem which is equivalent to a two-machine no-wait flow shop problem solvable in O(n log n) time. We then propose an optimal O(n log n) algorithm for the proportionate problem with equal machine speeds in which each job has the same processing time on both machines. We show that our O(n log n) algorithm also applies to the more general proportionate problem with equal machine speeds and machine-specific setup times. We also analyze the proportionate problem with unequal machine speeds and conclude that the required enumeration can be further curtailed (compared to the problem with arbitrary job processing times) by eliminating certain job pairs from consideration.     

Single-machine common flow allowance scheduling with job-dependent aging effects and a deteriorating maintenance activity

We consider a single machine due date assignment scheduling problem with job-dependent aging effects and a deteriorating maintenance activity, where due dates are assigned using the SLK due date determination method. We need to make a decision on when to schedule the deteriorating maintenance activity, the optimal common flow allowance and the sequence of jobs to minimize total earliness, tardiness and common flow allowance cost. We show that the problem remains polynomially solvable under the proposed model.     

Bi-criterion single-machine scheduling and due-window assignment with common flow allowances and resource-dependent processing times

This paper addresses single-machine scheduling and due-window assignment with common flow allowances and resource-dependent processing times. Due-window assignment with common flow allowances means that each job has a job-dependent due window, the start time and finish time of which are equal to its actual processing time plus individual job-independent parameters shared by all the jobs, respectively. The processing time of each job can be controlled by extra resource allocation as a linear function of the amount of a common continuously divisible resource allocated to the job. Two criteria are considered, where one criterion is an integrated cost consisting of job earliness, weighted number of tardy jobs, and due-window assignment cost, while the other criterion is the resource consumption cost. Four different models are considered for treating the two criteria. It is shown that the problem under the model where the two criteria are integrated into a single criterion is polynomially solvable, while the problems under the other three models are all NP-hard and an optimal solution procedure is developed for them. Two polynomially solvable cases are also identified and investigated. Finally, numerical studies with randomly generated instances are conducted to assess the performance of the proposed algorithms.     

A new exact discrete linear reformulation of the quadratic assignment problem

The quadratic assignment problem (QAP) is a challenging combinatorial problem. The problem is NP-hard and in addition, it is considered practically intractable to solve large QAP instances, to proven optimality, within reasonable time limits. In this paper we present an attractive mixed integer linear programming (MILP) formulation of the QAP. We first introduce a useful non-linear formulation of the problem and then a method of how to reformulate it to a new exact, compact discrete linear model. This reformulation is efficient for QAP instances with few unique elements in the flow or distance matrices. Finally, we present optimal results, obtained with the discrete linear reformulation, for some previously unsolved instances (with the size n = 32 and 64), from the quadratic assignment problem library, QAPLIB.     

Single machine scheduling with multiple common due date assignment and aging effect under a deteriorating maintenance activity consideration

In this paper, we consider the multiple common due date assignment and single machine scheduling with a job-dependent aging effect and a deteriorating maintenance activity. Once the maintenance activity has been completed, the machine will revert to its initial condition and the aging effect will start anew, the maintenance duration depends on its starting time. The objective is to minimize the total of earliness, tardiness, due date costs and find the optimal due date, the optimal maintenance position. We introduce an efficient O(n ⁴) algorithm to solve the problem. We also provide a special case of the problem and show that it remains polynomial time solvable.     

Optimal Assignment of Total-work-content Due-dates and Sequencing in a Single-machine Shop

This paper considers the problem of optimal assignment of total-work-content due-dates to n jobs and of sequencing them on a single machine to minimize an objective function depending on the assigned due-date multiple value and maximum tardiness penalty. It is shown that both the earliest due-date and shortest processing time orders yield an optimal sequence. A simple analytical solution method is presented to find the optimal due-dates. After the theoretical treatment an illustrative example is presented for discussion.     

Scheduling with controllable release dates and processing times: Makespan minimization

The paper deals with the single-machine scheduling problem in which job processing times as well as release dates are controllable parameters and they may vary within given intervals. While all release dates have the same boundary values, the processing time intervals are arbitrary. It is assumed that the cost of compressing processing times and release dates from their initial values is a linear function of the compression amount. The objective is to minimize the makespan together with the total compression cost. We construct a reduction to the assignment problem for the case of equal release date compression costs and develop an O(n²) algorithm for the case of equal release date compression costs and equal processing time compression costs. For the bicriteria version of the latter problem with agreeable processing times, we suggest an O(n²) algorithm that constructs the breakpoints of the efficient frontier.     

A polynomial algorithm for a two-machine no-wait job-shop scheduling problem

This paper considers the no-wait scheduling of n jobs, where each job is a chain of unit processing time operations to be processed alternately on two machines. The objective is to minimize the mean flow time. We propose an O(n⁶)-time algorithm to produce an optimal schedule. It is also shown that if zero processing time operations are allowed, then the problem is NP-hard in the strong sense.     

Minimizing the weighted number of tardy jobs with due date assignment and capacity-constrained deliveries for multiple customers in supply chains

In this paper, an integrated due date assignment and production and batch delivery scheduling problem for make-to-order production system and multiple customers is addressed. Consider a supply chain scheduling problem in which n orders (jobs) have to be scheduled on a single machine and delivered to K customers or to other machines for further processing in batches. A common due date is assigned to all the jobs of each customer and the number of jobs in delivery batches is constrained by the batch size. The objective is to minimize the sum of the total weighted number of tardy jobs, the total due date assignment costs and the total batch delivery costs. The problem is NP-hard. We formulate the problem as an Integer Programming (IP) model. Also, in this paper, a Heuristic Algorithm (HA) and a Branch and Bound (B&B) method for solving this problem are presented. Computational tests are used to demonstrate the efficiency of the developed methods.     

Balancing Workloads and Minimizing Set-up Costs in the Parallel Processing Shop

This paper presents an optimal scheduling algorithm for minimizing set-up costs in the parallel processing shop while meeting workload balancing restrictions.There are M independent batch type jobs which have sequence dependent set-up costs and N parallel processing machines. Each of the M jobs must be processed on exactly one of the N available machines. It is desirable to minimize total changeover costs with the restriction that each machine workload assignment T _n be within P units of the average machine assignment. The paper describes a static problem in which all jobs are available at time zero. The sequence dependent change over costs are identical for each machine. An extension of the algorithm handles nonidentical processor problems.A combinatorial programming approach to the problem is used. For the special case of identical processors, the problem can be treated as a multi-salesman travelling salesman problem. A general branch and bound algorithm and numerical results are given.     

Dynamic Capacity Acquisition and Assignment under Uncertainty

Given a set of m resources and n tasks, the dynamic capacity acquisition and assignment problem seeks a minimum cost schedule of capacity acquisitions for the resources and the assignment of resources to tasks, over a given planning horizon of T periods. This problem arises, for example, in the integrated planning of locations and capacities of distribution centers (DCs), and the assignment of customers to the DCs, in supply chain applications. We consider the dynamic capacity acquisition and assignment problem in an environment where the assignment costs and the processing requirements for the tasks are uncertain. Using a scenario based approach, we develop a stochastic integer programming model for this problem. The highly non-convex nature of this model prevents the application of standard stochastic programming decomposition algorithms. We use a recently developed decomposition based branch-and-bound strategy for the problem. Encouraging preliminary computational results are provided.     

Optimal due-date determination and sequencing with random processing times

This paper considers the problem of due-date determination and sequencing of n stochastically independent jobs on a single machine with random processing times. The objective is to find the optimal due-date values for the constant due-date assignment method and the optimal job sequence that minimize the expected value of a total cost function. It is shown that under suitable assumptions the optimal due-date values can be analytically determined and the jobs should be arranged in the SEPT sequence to minimize the cost.     

An assignment algorithm with applications to integrated circuit layout

The 2-terminal one-to-any problem, which arises in the design of layout systems, is the problem of assigning wach one of n terminals positioned on the upper row of a channel (called entry terminals) to one of m terminals positioned on the lower row (called exit terminals) so that the resulting channel routing problem has minimum density. An optimal solution to this problem is known [1]. In this paper we consider a natural generalization, the 2-color one-to-any problem, in which we have two types of entry terminals, red and blue ones, and exit terminals can be assigned to either type of entry terminal. Red and blue nets created by our algorithm are allowed to run on top of each other in the routing, and the density is defined as the larger of the red density and the blue density. Its minimization is an interesting combinatorial problem. We show how to compute the best achievable density in O(n + m) time, and an assignment achieving this density in O((n + m)log(n + m)) time.     

An O(n) algorithm for the variable common due date,minimal tardy jobs bicriteria two-machine flow shop problem with ordered machines

We consider the ordinary NP- hard two-machine flow shop problem with the objective of determining simultaneously a minimal common due date and the minimal number of tardy jobs. We present an O(n²) algorithm for the problem when the machines are ordered, that is, when each job has its smaller processing time on the first (second) machine. We also discuss the applicability of the proposed algorithm to the corresponding single-objective problem in which the common due date is given.