Assembly line balancing problems with imprecise input data (durations of assembly operations, num... more Assembly line balancing problems with imprecise input data (durations of assembly operations, number of workstations) are considered. Problem settings with deterministic, stochastic, and uncertain parameters are discussed. Different approaches to the assembly line balancing problems with imprecise durations of assembly operations are analyzed. A new problem setting is proposed in which durations of assembly operations are given by lower and upper bounds of their possible values.
We propose the adaptive algorithm for solving a set of similar scheduling problems using learning... more We propose the adaptive algorithm for solving a set of similar scheduling problems using learning technology. It is devised to combine the merits of an exact algorithm based on the mixed graph model and heuristics oriented on the real-world scheduling problems. The former may ensure high quality of the solution by means of an implicit exhausting enumeration of the feasible schedules. The latter may be developed for certain type of problems using their peculiarities. The main idea of the learning technology is to produce effective (in performance measure) and efficient (in computational time) heuristics by adapting local decisions for the scheduling problems under consideration. Adaptation is realized at the stage of learning while solving a set of sample scheduling problems using a branch-and-bound algorithm and structuring knowledge using pattern recognition apparatus.
This paper addresses the two-machine flowshop scheduling problem with separate setup times to min... more This paper addresses the two-machine flowshop scheduling problem with separate setup times to minimize makespan or total completion time (TCT). Setup times are relaxed to be random variables rather than deterministic as commonly used in the OR literature. Moreover, distribution-free setup times are used where only the lower and upper bounds are given. Global and local dominance relations are developed for the considered flowshops and an illustrative numerical example is given. 2000 Mathematics Subject Classification: 90B36, 90B35, 90B30. 1. Introduction. The
A single-machine scheduling problem is investigated provided that the input data are uncertain: T... more A single-machine scheduling problem is investigated provided that the input data are uncertain: The processing time of a job can take any real value from the given segment. The criterion is to minimize the total weighted completion time for the n jobs. As a solution concept to such a scheduling problem with an uncertain input data, it is reasonable to consider a minimal dominant set of job permutations containing an optimal permutation for each possible realization of the job processing times. To find an optimal or approximate job permutation to be realized, we look for a permutation with the largest stability box being a subset of the stability region. We develop a branch-and-bound algorithm to construct a permutation with the largest volume of a stability box. If several permutations have the same volume of a stability box, we select one of them due to one of two simple heuristics. The efficiency of the constructed permutations (how close they are to a factually optimal permutatio...
Stable optimal line balances with a fixed station set
A simple assembly line balancing problem is considered provided that number m of working stations... more A simple assembly line balancing problem is considered provided that number m of working stations is xed. In such a problem, denoted as SALBP-2, it is necessary to minimize a cycle time for processing a partially ordered set of assembly operations V = f 1 ; 2 ;:::;n g on a set of m linearly ordered working stations. An initial vector of the processing times t = ( t 1 ;t 2 ;:::;t n ) of the (assembly) operations V is known before solving the problem SALBP-2. For each an automated operation i 2 V n e V , the processing time t i cannot vary during a life cycle of the assembly line. For a subset e V V of the manual operations j 2 e V , the processing times t j may vary, since di erent operators (workers) may have di erent skill, experience, etc. We investigate a stability of the optimal line balance of the simple assembly line with respect to simultaneous variations of the processing times t j of the manual operations j 2 e V . An optimal line balance is stable if its optimality is pres...
A Stability Approach to Sequencing and Scheduling under Uncertainty
An early version of this chapter is downlodable at ResearchGate as Preprint 09/13, FMA, OvGU Magd... more An early version of this chapter is downlodable at ResearchGate as Preprint 09/13, FMA, OvGU Magdeburg, June 2013, under the title 'Using a Stability Method for Scheduling and Sequencing with Interval Data'
In project management, it is usually difficult to obtain the exact values of the activity duratio... more In project management, it is usually difficult to obtain the exact values of the activity durations and the assumption is more realistic that the activity duration may remain uncertain until the activity completion. We assume that lower and upper bounds on a factual activity duration are given at the stage of project planning, the probability distribution of a random duration being unknown before the activity completion. Therefore, one cannot find a priory a critical path in the given project-network G. We propose a two-step approach, where the initial project-network G is minimized in the first step and the resulting minimized project-network determines a minimal dominant set of the critical paths in the second step. A fuzzy logic procedure (or another heuristic technique) may be used to choose a single potentially critical path from the minimal dominant set.
The main objective of this paper is to stimulate interest in stability analysis for scheduling pr... more The main objective of this paper is to stimulate interest in stability analysis for scheduling problems. In spite of impressive theoretical results in sequencing and scheduling, up to now the implementation of scheduling algorithms with a rather deep mathematical background in production planning, scheduling and control, and in other real-life problems with sequencing aspects is limited. In classical scheduling theory, mainly deterministic systems are considered and the processing times of all operations are supposed to be given in advance. Such problems do not often arise in practice: Even if the processing times are known before applying a scheduling procedure, OR workers are forced to take i n to account the precision of equipment, which is used to calculate the processing times, round-o errors in the calculation of a schedule, errors within the practical realization of a schedule, machine breakdowns, additional jobs and so on. This paper is devoted to the calculation of the stability radius of an optimal or an approximate schedule. We survey some recent results in this eld and derive new results in order to make this approach more suitable for practical use. Computational results on the calculation of the stability radius for randomly generated job shop scheduling problems are presented. The extreme values of the stability radius are considered in more detail. The new results are amply illustrated with examples.
The usual assumption that the processing times of the operations are known in advance is the stri... more The usual assumption that the processing times of the operations are known in advance is the strictest one in deterministic scheduling theory and it essentially restricts its practical aspects. Indeed, this assumption is not valid for the most real-world processes. This paper is devoted to a stability analysis of an optimal schedule which may help to extend the significance of scheduling theory for some production scheduling problems. The terms 'stability', 'sensitivity' or 'postoptimal analysis' are generally used for the phase of an algorithm at which a solution (or solutions) of an optimization problem has already been found, and additional calculations are performed in order to investigate how this solution depends on the problem data. We survey some recent results in the calculation of the stability radius of an optimal schedule for a general shop scheduling problem which denotes the largest quantity of independent variations of the processing times of the operations such that this schedule remains optimal. We present formulas for the calculation of the stability radius, when the objective is to minimize mean or maximum flow time. The extreme values of the stability radius are of particular importance, and these cases are considered more in detail. Moreover, computational results on the calculation of the stability radius for randomly generated job shop scheduling problems are briefly discussed. The developed software allows us to show that the most optimal schedules of the well-known test problem with 6 jobs and 6 machines are unstable.
A learning stage of scheduling tends to produce knowledge about a benchmark of priority dispatchi... more A learning stage of scheduling tends to produce knowledge about a benchmark of priority dispatching rules which allows a scheduler to improve the solution quality for a set of similar job-shop problems. Once trained on the sample job-shop problems (usually with small sizes), the adaptive algorithm solves a similar job-shop problem (with a moderate size or a large size) better than heuristics used as a benchmark at the learning stage of scheduling. Our adaptive algorithm does not guarantee to perform as an exact algorithm or better than a more sophisticated heuristic algorithm (like e.g. the shifting bottleneck one) which need a large running time. For an adaptive algorithm with a learning stage, the job-shop scheduling problem is modeled via a weighted mixed (disjunctive) graph with the conflict resolution strategy used for finding an appropriate schedule.
We consider a job shop scheduling problem under uncertain processing times and fixed precedence a... more We consider a job shop scheduling problem under uncertain processing times and fixed precedence and capacity constraints. Each of the random processing times can take any real value between given lower and upper bounds. The goal is to find a set of schedules which contains at least one optimal schedule (with mean flow time criterion) for any admissible realization of the random processing times. In order to compute such a set of schedules efficiently and keep it as small as possible, we develop several exact and heuristic algorithms and report computational experience based on randomly generated instances.
Some scheduling problems induce a mixed graph coloring, i.e. an assignment of positive i n tegers... more Some scheduling problems induce a mixed graph coloring, i.e. an assignment of positive i n tegers (colors) to vertices of a mixed graph such that, if two v ertices are joined by an edge, then their colors have to be di erent, and if two v ertices are joined by an arc, then the color of the startvertex has to be not greater than the color of the endvertex. We discuss some algorithms for coloring the vertices of a mixed graph with a small numbert of colors and present computational results for calculating the chromatic number, i.e. the minimal possible value of such a t. We also study the chromatic polynomial of a mixed graph which may b e used for calculating the numberof feasible schedules.
We consider a job shop scheduling problem under uncertain processing times and fixed precedence a... more We consider a job shop scheduling problem under uncertain processing times and fixed precedence and capacity constraints. Each of the random processing times can take any real value between given lower and upper bounds. The goal is to find a set of schedules which contains at least one optimal schedule (with mean flow time criterion) for any admissible realization of the random processing times. In order to compute such a set of schedules efficiently and keep it as small as possible, we develop several exact and heuristic algorithms and report computational experience based on randomly generated instances.
Complexity of mixed shop scheduling problems: A survey
European Journal of Operational Research, 2000
We survey recent results on the computational complexity of mixed shop scheduling problems. In a ... more We survey recent results on the computational complexity of mixed shop scheduling problems. In a mixed shop, some jobs have fixed machine orders (as in the job shop), while the operations of the other jobs may be processed in arbitrary order (as in the open shop). The main attention is devoted to establishing the boundary between polynomially solvable and NP-hard problems. When the number of operations per job is unlimited, we focus on problems with a fixed number of jobs.
Your article is protected by copyright and all rights are held exclusively by Springer Science+Bu... more Your article is protected by copyright and all rights are held exclusively by Springer Science+Business Media, LLC. This e-offprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to self-archive your work, please use the accepted author's version for posting to your own website or your institution's repository. You may further deposit the accepted author's version on a funder's repository at a funder's request, provided it is not made publicly available until 12 months after publication.
A set J = {J1, J2, . . . , Jn} of jobs has to be processed on a set of parallel uniform machines.... more A set J = {J1, J2, . . . , Jn} of jobs has to be processed on a set of parallel uniform machines. For each job Ji ∈ J , a release time ri ≥ 0 and a due date di > ri are given. Each job may be processed without interruptions on any of the given machines having different speeds. If job Ji ∈ J will be started and then completed within the time segment [ri, di], the benefit bi > 0 will be earned. Otherwise, this job will be rejected and the benefit bi will be discarded. Let J (S) denote the subset of all jobs Ji ∈ J processed within their intervals [ri, di] in the schedule S. The set J \ J (S) includes all the jobs rejected from the schedule S. The criterion under consideration is to maximize the weighted sum of the benefits w1 ∑ i∈J (S) bi and the weighted number of jobs w2|J (S)| processed according to the schedule S, where both weights w1 and w2 are non-negative with the assumption that w1 + w2 = 1. We investigate some properties of the objective function, develop a simulated a...
In practice, it is often required to process a set of jobs without operation preemptions satisfyi... more In practice, it is often required to process a set of jobs without operation preemptions satisfying temporal and resource constraints. Temporal constraints say that some jobs have to be finished before some others can be started. Resource constraints say that operations processed on the same machine cannot be processed simultaneously. The objective is to construct a schedule specifying when each operation starts such that both temporal and resource constraints are satisfied and the given objective function has a minimum value. One can model such a scheduling process via the following job-shop problem. There are n jobs J = {J1, J2, . . . , Jn} to be processed on m machines M = {M1,M2, . . . ,Mm}. Operation preemptions are not allowed, and the machine routes O = (Oi1, Oi2, . . . , Oini) for processing the jobs Ji ∈ J may be given differently for different jobs. The time pij > 0 needed for processing an operation Oij of a job Ji on the corresponding machine Mv ∈ M is known before sc...
Assembly line balancing problems with imprecise input data (durations of assembly operations, num... more Assembly line balancing problems with imprecise input data (durations of assembly operations, number of workstations) are considered. Problem settings with deterministic, stochastic, and uncertain parameters are discussed. Different approaches to the assembly line balancing problems with imprecise durations of assembly operations are analyzed. A new problem setting is proposed in which durations of assembly operations are given by lower and upper bounds of their possible values.
We propose the adaptive algorithm for solving a set of similar scheduling problems using learning... more We propose the adaptive algorithm for solving a set of similar scheduling problems using learning technology. It is devised to combine the merits of an exact algorithm based on the mixed graph model and heuristics oriented on the real-world scheduling problems. The former may ensure high quality of the solution by means of an implicit exhausting enumeration of the feasible schedules. The latter may be developed for certain type of problems using their peculiarities. The main idea of the learning technology is to produce effective (in performance measure) and efficient (in computational time) heuristics by adapting local decisions for the scheduling problems under consideration. Adaptation is realized at the stage of learning while solving a set of sample scheduling problems using a branch-and-bound algorithm and structuring knowledge using pattern recognition apparatus.
This paper addresses the two-machine flowshop scheduling problem with separate setup times to min... more This paper addresses the two-machine flowshop scheduling problem with separate setup times to minimize makespan or total completion time (TCT). Setup times are relaxed to be random variables rather than deterministic as commonly used in the OR literature. Moreover, distribution-free setup times are used where only the lower and upper bounds are given. Global and local dominance relations are developed for the considered flowshops and an illustrative numerical example is given. 2000 Mathematics Subject Classification: 90B36, 90B35, 90B30. 1. Introduction. The
A single-machine scheduling problem is investigated provided that the input data are uncertain: T... more A single-machine scheduling problem is investigated provided that the input data are uncertain: The processing time of a job can take any real value from the given segment. The criterion is to minimize the total weighted completion time for the n jobs. As a solution concept to such a scheduling problem with an uncertain input data, it is reasonable to consider a minimal dominant set of job permutations containing an optimal permutation for each possible realization of the job processing times. To find an optimal or approximate job permutation to be realized, we look for a permutation with the largest stability box being a subset of the stability region. We develop a branch-and-bound algorithm to construct a permutation with the largest volume of a stability box. If several permutations have the same volume of a stability box, we select one of them due to one of two simple heuristics. The efficiency of the constructed permutations (how close they are to a factually optimal permutatio...
Stable optimal line balances with a fixed station set
A simple assembly line balancing problem is considered provided that number m of working stations... more A simple assembly line balancing problem is considered provided that number m of working stations is xed. In such a problem, denoted as SALBP-2, it is necessary to minimize a cycle time for processing a partially ordered set of assembly operations V = f 1 ; 2 ;:::;n g on a set of m linearly ordered working stations. An initial vector of the processing times t = ( t 1 ;t 2 ;:::;t n ) of the (assembly) operations V is known before solving the problem SALBP-2. For each an automated operation i 2 V n e V , the processing time t i cannot vary during a life cycle of the assembly line. For a subset e V V of the manual operations j 2 e V , the processing times t j may vary, since di erent operators (workers) may have di erent skill, experience, etc. We investigate a stability of the optimal line balance of the simple assembly line with respect to simultaneous variations of the processing times t j of the manual operations j 2 e V . An optimal line balance is stable if its optimality is pres...
A Stability Approach to Sequencing and Scheduling under Uncertainty
An early version of this chapter is downlodable at ResearchGate as Preprint 09/13, FMA, OvGU Magd... more An early version of this chapter is downlodable at ResearchGate as Preprint 09/13, FMA, OvGU Magdeburg, June 2013, under the title 'Using a Stability Method for Scheduling and Sequencing with Interval Data'
In project management, it is usually difficult to obtain the exact values of the activity duratio... more In project management, it is usually difficult to obtain the exact values of the activity durations and the assumption is more realistic that the activity duration may remain uncertain until the activity completion. We assume that lower and upper bounds on a factual activity duration are given at the stage of project planning, the probability distribution of a random duration being unknown before the activity completion. Therefore, one cannot find a priory a critical path in the given project-network G. We propose a two-step approach, where the initial project-network G is minimized in the first step and the resulting minimized project-network determines a minimal dominant set of the critical paths in the second step. A fuzzy logic procedure (or another heuristic technique) may be used to choose a single potentially critical path from the minimal dominant set.
The main objective of this paper is to stimulate interest in stability analysis for scheduling pr... more The main objective of this paper is to stimulate interest in stability analysis for scheduling problems. In spite of impressive theoretical results in sequencing and scheduling, up to now the implementation of scheduling algorithms with a rather deep mathematical background in production planning, scheduling and control, and in other real-life problems with sequencing aspects is limited. In classical scheduling theory, mainly deterministic systems are considered and the processing times of all operations are supposed to be given in advance. Such problems do not often arise in practice: Even if the processing times are known before applying a scheduling procedure, OR workers are forced to take i n to account the precision of equipment, which is used to calculate the processing times, round-o errors in the calculation of a schedule, errors within the practical realization of a schedule, machine breakdowns, additional jobs and so on. This paper is devoted to the calculation of the stability radius of an optimal or an approximate schedule. We survey some recent results in this eld and derive new results in order to make this approach more suitable for practical use. Computational results on the calculation of the stability radius for randomly generated job shop scheduling problems are presented. The extreme values of the stability radius are considered in more detail. The new results are amply illustrated with examples.
The usual assumption that the processing times of the operations are known in advance is the stri... more The usual assumption that the processing times of the operations are known in advance is the strictest one in deterministic scheduling theory and it essentially restricts its practical aspects. Indeed, this assumption is not valid for the most real-world processes. This paper is devoted to a stability analysis of an optimal schedule which may help to extend the significance of scheduling theory for some production scheduling problems. The terms 'stability', 'sensitivity' or 'postoptimal analysis' are generally used for the phase of an algorithm at which a solution (or solutions) of an optimization problem has already been found, and additional calculations are performed in order to investigate how this solution depends on the problem data. We survey some recent results in the calculation of the stability radius of an optimal schedule for a general shop scheduling problem which denotes the largest quantity of independent variations of the processing times of the operations such that this schedule remains optimal. We present formulas for the calculation of the stability radius, when the objective is to minimize mean or maximum flow time. The extreme values of the stability radius are of particular importance, and these cases are considered more in detail. Moreover, computational results on the calculation of the stability radius for randomly generated job shop scheduling problems are briefly discussed. The developed software allows us to show that the most optimal schedules of the well-known test problem with 6 jobs and 6 machines are unstable.
A learning stage of scheduling tends to produce knowledge about a benchmark of priority dispatchi... more A learning stage of scheduling tends to produce knowledge about a benchmark of priority dispatching rules which allows a scheduler to improve the solution quality for a set of similar job-shop problems. Once trained on the sample job-shop problems (usually with small sizes), the adaptive algorithm solves a similar job-shop problem (with a moderate size or a large size) better than heuristics used as a benchmark at the learning stage of scheduling. Our adaptive algorithm does not guarantee to perform as an exact algorithm or better than a more sophisticated heuristic algorithm (like e.g. the shifting bottleneck one) which need a large running time. For an adaptive algorithm with a learning stage, the job-shop scheduling problem is modeled via a weighted mixed (disjunctive) graph with the conflict resolution strategy used for finding an appropriate schedule.
We consider a job shop scheduling problem under uncertain processing times and fixed precedence a... more We consider a job shop scheduling problem under uncertain processing times and fixed precedence and capacity constraints. Each of the random processing times can take any real value between given lower and upper bounds. The goal is to find a set of schedules which contains at least one optimal schedule (with mean flow time criterion) for any admissible realization of the random processing times. In order to compute such a set of schedules efficiently and keep it as small as possible, we develop several exact and heuristic algorithms and report computational experience based on randomly generated instances.
Some scheduling problems induce a mixed graph coloring, i.e. an assignment of positive i n tegers... more Some scheduling problems induce a mixed graph coloring, i.e. an assignment of positive i n tegers (colors) to vertices of a mixed graph such that, if two v ertices are joined by an edge, then their colors have to be di erent, and if two v ertices are joined by an arc, then the color of the startvertex has to be not greater than the color of the endvertex. We discuss some algorithms for coloring the vertices of a mixed graph with a small numbert of colors and present computational results for calculating the chromatic number, i.e. the minimal possible value of such a t. We also study the chromatic polynomial of a mixed graph which may b e used for calculating the numberof feasible schedules.
We consider a job shop scheduling problem under uncertain processing times and fixed precedence a... more We consider a job shop scheduling problem under uncertain processing times and fixed precedence and capacity constraints. Each of the random processing times can take any real value between given lower and upper bounds. The goal is to find a set of schedules which contains at least one optimal schedule (with mean flow time criterion) for any admissible realization of the random processing times. In order to compute such a set of schedules efficiently and keep it as small as possible, we develop several exact and heuristic algorithms and report computational experience based on randomly generated instances.
Complexity of mixed shop scheduling problems: A survey
European Journal of Operational Research, 2000
We survey recent results on the computational complexity of mixed shop scheduling problems. In a ... more We survey recent results on the computational complexity of mixed shop scheduling problems. In a mixed shop, some jobs have fixed machine orders (as in the job shop), while the operations of the other jobs may be processed in arbitrary order (as in the open shop). The main attention is devoted to establishing the boundary between polynomially solvable and NP-hard problems. When the number of operations per job is unlimited, we focus on problems with a fixed number of jobs.
Your article is protected by copyright and all rights are held exclusively by Springer Science+Bu... more Your article is protected by copyright and all rights are held exclusively by Springer Science+Business Media, LLC. This e-offprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to self-archive your work, please use the accepted author's version for posting to your own website or your institution's repository. You may further deposit the accepted author's version on a funder's repository at a funder's request, provided it is not made publicly available until 12 months after publication.
A set J = {J1, J2, . . . , Jn} of jobs has to be processed on a set of parallel uniform machines.... more A set J = {J1, J2, . . . , Jn} of jobs has to be processed on a set of parallel uniform machines. For each job Ji ∈ J , a release time ri ≥ 0 and a due date di > ri are given. Each job may be processed without interruptions on any of the given machines having different speeds. If job Ji ∈ J will be started and then completed within the time segment [ri, di], the benefit bi > 0 will be earned. Otherwise, this job will be rejected and the benefit bi will be discarded. Let J (S) denote the subset of all jobs Ji ∈ J processed within their intervals [ri, di] in the schedule S. The set J \ J (S) includes all the jobs rejected from the schedule S. The criterion under consideration is to maximize the weighted sum of the benefits w1 ∑ i∈J (S) bi and the weighted number of jobs w2|J (S)| processed according to the schedule S, where both weights w1 and w2 are non-negative with the assumption that w1 + w2 = 1. We investigate some properties of the objective function, develop a simulated a...
In practice, it is often required to process a set of jobs without operation preemptions satisfyi... more In practice, it is often required to process a set of jobs without operation preemptions satisfying temporal and resource constraints. Temporal constraints say that some jobs have to be finished before some others can be started. Resource constraints say that operations processed on the same machine cannot be processed simultaneously. The objective is to construct a schedule specifying when each operation starts such that both temporal and resource constraints are satisfied and the given objective function has a minimum value. One can model such a scheduling process via the following job-shop problem. There are n jobs J = {J1, J2, . . . , Jn} to be processed on m machines M = {M1,M2, . . . ,Mm}. Operation preemptions are not allowed, and the machine routes O = (Oi1, Oi2, . . . , Oini) for processing the jobs Ji ∈ J may be given differently for different jobs. The time pij > 0 needed for processing an operation Oij of a job Ji on the corresponding machine Mv ∈ M is known before sc...
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