## scheduling theory algorithms and systems solution 15

<br> - What are the main types of scheduling problems and models? <br> - What are some of the challenges and applications of scheduling theory? H2: Deterministic Scheduling Models - What are deterministic scheduling models and what are their assumptions? <br> - What are some of the common performance measures and objectives in deterministic scheduling? <br> - What are some of the basic deterministic scheduling models and how to solve them? <br> - Single machine models <br> - Parallel machine models <br> - Flow shop models <br> - Job shop models <br> - Open shop models H3: Stochastic Scheduling Models - What are stochastic scheduling models and what are their assumptions? <br> - How to deal with uncertainty and randomness in scheduling? <br> - What are some of the common performance measures and objectives in stochastic scheduling? <br> - What are some of the basic stochastic scheduling models and how to solve them? <br> - Single machine models with random processing times or release dates <br> - Parallel machine models with random processing times or release dates <br> - Flow shop, job shop, and open shop models with random processing times or release dates H4: Scheduling in Practice - What are some of the general purpose procedures for deterministic and stochastic scheduling? <br> - How to model and solve scheduling problems in practice using mathematical programming, heuristics, metaheuristics, simulation, etc.? <br> - What are some of the design and implementation issues of scheduling systems? <br> - How to evaluate and compare different scheduling methods and solutions? H5: Conclusion - Summarize the main points of the article <br> - Highlight the current trends and future directions of scheduling theory research and practice H6: FAQs - Q1: What is the difference between online and offline scheduling? <br> - Q2: What is the difference between preemptive and non-preemptive scheduling? <br> - Q3: What is the difference between makespan and flowtime? <br> - Q4: What is the difference between NP-hard and NP-complete problems? <br> - Q5: What are some of the real-world examples of scheduling problems? Article with HTML formatting <h1>Introduction</h1>

<p>Scheduling is the process of allocating scarce resources (such as machines, workers, tools, etc.) to a set of tasks (such as jobs, operations, activities, etc.) over time. Scheduling theory is the branch of operations research that studies how to design efficient and effective schedules that optimize one or more performance measures (such as completion time, cost, quality, etc.). Scheduling theory has many applications in various domains such as manufacturing, transportation, health care, education, project management, etc.</p>

<p>Scheduling problems can be classified into different types according to their characteristics and assumptions. Some of the main types are:</p>

<ul>

<li>Deterministic scheduling problems: These are problems where all the relevant data (such as processing times, due dates, precedence relations, etc.) are known in advance and do not change over time.</li>

<li>Stochastic scheduling problems: These are problems where some or all of the relevant data are uncertain or random and may change over time.</li>

<li>Online scheduling problems: These are problems where the tasks arrive dynamically over time and the scheduler has to make decisions without knowing the future arrivals.</li>

<li>Offline scheduling problems: These are problems where all the tasks are known in advance and the scheduler can plan ahead.</li>

<li>Preemptive scheduling problems: These are problems where a task can be interrupted and resumed later without any penalty.</li>

<li>Non-preemptive scheduling problems: These are problems where a task cannot be interrupted once it starts.</li>

</ul>

<p>Scheduling theory is a challenging and active research area that involves many mathematical models, algorithms, heuristics, simulation techniques, computational complexity analysis, etc. In this article, we will review some of the basic concepts and methods of scheduling theory, focusing on deterministic, stochastic, and practical scheduling models and problems. We will also discuss some of the current trends and future directions of scheduling theory research and practice.</p>

<h2>Deterministic Scheduling Models</h2>

<p>Deterministic scheduling models are models where all the relevant data are known in advance and do not change over time. These models are based on the following assumptions:</p>

<ul>

<li>The set of tasks and the set of resources are finite and fixed.</li>

<li>The processing time of each task on each resource is constant and independent of other tasks and resources.</li>

<li>The due date, priority, weight, or other attributes of each task are constant and independent of other tasks.</li>

<li>The precedence relations, resource constraints, or other restrictions among the tasks are constant and independent of time.</li>

</ul>

<p>Deterministic scheduling models aim to optimize one or more performance measures that reflect the quality of the schedule. Some of the common performance measures and objectives are:</p>

<ul>

<li>Makespan: The total completion time of all the tasks.</li>

<li>Flowtime: The sum of the completion times of all the tasks.</li>

<li>Lateness: The difference between the completion time and the due date of a task.</li>

<li>Tardiness: The positive part of the lateness of a task.</li>

<li>Earliness: The negative part of the lateness of a task.</li>

<li>Number of tardy tasks: The number of tasks that have positive tardiness.</li>

<li>Total weighted tardiness: The sum of the products of the weight and the tardiness of each task.</li>

<li>Resource utilization: The ratio of the busy time to the total time of a resource.</li>

</ul>

<p>Deterministic scheduling models can be further classified into different types according to the number and type of resources and tasks. Some of the basic deterministic scheduling models and how to solve them are:</p>

<h3>Single machine models</h3>

<p>Single machine models are models where there is only one resource (a machine) that can process one task at a time. The tasks are independent and have no precedence relations. The objective is to find a sequence of tasks that minimizes a performance measure. Some examples of single machine models and their optimal solutions are:</p>

<ul>

<li>Minimizing makespan: This is equivalent to minimizing the completion time of the last task. The optimal solution is any sequence that satisfies the precedence relations (if any).</li>

<li>Minimizing flowtime: This is equivalent to minimizing the average completion time of all the tasks. The optimal solution is the shortest processing time (SPT) rule, which sequences the tasks in non-decreasing order of their processing times.</li>

<li>Minimizing total weighted flowtime: This is equivalent to minimizing the weighted average completion time of all the tasks. The optimal solution is the weighted shortest processing time (WSPT) rule, which sequences the tasks in non-increasing order of their ratios of weight to processing time.</li>

<li>Minimizing maximum lateness: This is equivalent to minimizing the maximum difference between the completion time and the due date of any task. The optimal solution is the earliest due date (EDD) rule, which sequences the tasks in non-decreasing order of their due dates.</li>

<li>Minimizing number of tardy tasks: This is equivalent to minimizing the number of tasks that have positive lateness. The optimal solution is given by an algorithm called Moore's algorithm, which first applies the EDD rule and then eliminates any tardy tasks from the end of the sequence until no tardy task remains.</li>

<li>Minimizing total weighted tardiness: This is equivalent to minimizing the sum of the products of the weight and the tardiness of each task. The optimal solution is given by an algorithm called Lawler's algorithm, which is based on dynamic programming and branch-and-bound techniques.</li>

</ul>

<h3>Parallel machine models</h3>

<p>Parallel machine models are models where there are multiple identical resources (machines) that can process one task at a time. The tasks are independent and have no precedence relations. The objective is to find an assignment of tasks to machines and a sequence of tasks on each machine that minimizes a performance measure. Some examples of parallel machine models and their optimal or approximate solutions are:</p>

<ul>

<li>Minimizing makespan: This is equivalent to minimizing the maximum completion time among all the machines. The optimal solution is given by an algorithm called LPT (longest processing time) rule, which assigns the tasks in non-increasing order of their processing times to the machine with the smallest load so far.</li>

<h3>Flow shop models</h3>

<p>Flow shop models are models where there are multiple resources (machines) arranged in a series. Each task (job) consists of multiple operations that have to be processed on the machines in the same order. The objective is to find a sequence of tasks and a schedule of operations on each machine that minimizes a performance measure. Some examples of flow shop models and their optimal or approximate solutions are:</p>

<ul>

<li>Minimizing makespan: This is equivalent to minimizing the completion time of the last operation of the last task. The optimal solution is given by an algorithm called Johnson's rule, which applies to the case where there are two machines and each operation has a constant processing time on each machine. The algorithm sorts the tasks into two groups according to whether their first operation is shorter or longer than their second operation, and then sequences the tasks in non-decreasing order of their first operation in the first group and in non-increasing order of their second operation in the second group.</li>

<li>Minimizing flowtime: This is equivalent to minimizing the sum or average completion time among all the operations of all the tasks. The optimal solution is NP-hard even for two machines. A common heuristic is the shortest processing time (SPT) rule, which sequences the tasks in non-decreasing order of their total processing time on all machines.</li>

<li>Minimizing total weighted flowtime: This is equivalent to minimizing the weighted sum or average completion time among all the operations of all the tasks. The optimal solution is NP-hard even for two machines. A common heuristic is the weighted shortest processing time (WSPT) rule, which sequences the tasks in non-increasing order of their ratios of weight to total processing time on all machines.</li>

</ul>

<h3>Job shop models</h3>

<p>Job shop models are models where there are multiple resources (machines) that can process different types of operations. Each task (job) consists of multiple operations that have to be processed on different machines in a specific order. The objective is to find a schedule of operations on each machine that minimizes a performance measure. Some examples of job shop models and their optimal or approximate solutions are:</p>

<ul>

<li>Minimizing makespan: This is equivalent to minimizing the completion time of the last operation of any task. The optimal solution is NP-hard even for two machines and two tasks. A common heuristic is the shortest processing time first (SPTF) rule, which assigns the operation with the shortest processing time among all available operations to the machine with the smallest load so far.</li>

<li>Minimizing flowtime: This is equivalent to minimizing the sum or average completion time among all the operations of all the tasks. The optimal solution is NP-hard even for two machines and three tasks. A common heuristic is the earliest due date first (EDDF) rule, which assigns the operation with the earliest due date among all available operations to the machine with the smallest load so far.</li>

<li>Minimizing total weighted tardiness: This is equivalent to minimizing the sum of the products of the weight and the tardiness of each operation. The optimal solution is NP-hard even for two machines and three tasks. A common heuristic is a genetic algorithm that uses a permutation representation with repetition for encoding schedules and various operators for generating and improving solutions.</li>

</ul>

<h3>Open shop models</h3>

<p>Open shop models are models where there are multiple resources (machines) that can process different types of operations. Each task (job) consists of multiple operations that have to be processed on different machines in any order. The objective is to find a schedule of operations on each machine that minimizes a performance measure. Some examples of open shop models and their optimal or approximate solutions are:</p>

<ul>

<li>Minimizing makespan: This is equivalent to minimizing the completion time of any operation of any task. The optimal solution is NP-hard even for two machines and three tasks. A common heuristic is a greedy algorithm that assigns an operation with maximum processing time among all available operations to a machine with minimum load so far.</li>

<h3>Open shop models</h3>

<p>Open shop models are models where there are multiple resources (machines) that can process different types of operations. Each task (job) consists of multiple operations that have to be processed on different machines in any order. The objective is to find a schedule of operations on each machine that minimizes a performance measure. Some examples of open shop models and their optimal or approximate solutions are:</p>

<ul>

<li>Minimizing makespan: This is equivalent to minimizing the completion time of any operation of any task. The optimal solution is NP-hard even for two machines and three tasks. A common heuristic is a greedy algorithm that assigns an operation with maximum processing time among all available operations to a machine with minimum load so far.</li>

<li>Minimizing flowtime: This is equivalent to minimizing the sum or average completion time among all the operations of all the tasks. The optimal solution is NP-hard even for two machines and four tasks. A common heuristic is a genetic algorithm that uses a permutation representation with repetition for encoding schedules and various operators for generating and improving solutions.</li>

<li>Minimizing total weighted tardiness: This is equivalent to minimizing the sum of the products of the weight and the tardiness of each operation. The optimal solution is NP-hard even for two machines and four tasks. A common heuristic is a simulated annealing algorithm that uses a permutation representation with repetition for encoding schedules and various operators for generating and improving solutions.</li>

</ul>

<h4>Scheduling in Practice</h4>

<p>Scheduling in practice involves dealing with more complex and realistic scheduling problems that may not fit into the standard models and assumptions discussed above. Some of the challenges and issues that arise in scheduling in practice are:</p>

<ul>

<li>The data may be incomplete, uncertain, or dynamic, requiring online, adaptive, or robust scheduling methods.</li>

<li>The objectives may be multiple, conflicting, or fuzzy, requiring multi-criteria, game-theoretic, or fuzzy logic approaches.</li>

<li>The constraints may be more diverse, such as setup times, setup resources, batching, precedence relations, release dates, due dates, deadlines, time windows, etc., requiring more sophisticated modeling and solving techniques.</li>

<li>The problems may be too large or complex to be solved optimally or exactly, requiring approximation, heuristic, metaheuristic, or hybrid methods.</li>

<li>The solutions may need to be evaluated and compared using various criteria, such as optimality gap, computational time, solution quality, robustness, stability, etc., requiring performance analysis and benchmarking tools.</li>

<li>The solutions may need to be implemented and executed in real systems, requiring system design and integration issues such as user interface, data input/output, communication protocol, feedback mechanism, etc.</li>

</ul>

<p>Some of the general purpose procedures for deterministic and stochastic scheduling are:</p>

<ul>

<li>Mathematical programming: This is a method that formulates the scheduling problem as an optimization problem using variables, constraints, and objective functions. The optimization problem can be solved using various techniques such as linear programming (LP), integer programming (IP), mixed-integer programming (MIP), nonlinear programming (NLP), etc.</li>

<li>Heuristics: These are methods that use simple rules or strategies to generate feasible or near-optimal solutions quickly without guaranteeing optimality or quality. Some examples of heuristics are priority rules (such as SPT, EDD, WSPT), dispatching rules (such as first come first served (FCFS), shortest queue (SQ), longest idle time (LIT)), etc.</li>

<h4>Scheduling in Practice</h4>

<p>Scheduling in practice involves dealing with more complex and realistic scheduling problems that may not fit into the standard models and assumptions discussed above. Some of the challenges and issues that arise in scheduling in practice are:</p>

<ul>

<li>The data may be incomplete, uncertain, or dynamic, requiring online, adaptive, or robust scheduling methods.</li>

<li>The objectives may be multiple, conflicting, or fuzzy, requiring multi-criteria, game-theoretic, or fuzzy logic approaches.</li>

<li>The constraints may be more diverse, such as setup times, setup resources, batching, precedence relations, release dates, due dates, deadlines, time windows, etc., requiring more sophisticated modeling and solving techniques.</li>

<li>The problems may be too large or complex to be solved optimally or exactly, requiring approximation, heuristic, metaheuristic, or hybrid methods.</li>

<li>The solutions may need to be evaluated and compared using various criteria, such as optimality gap, computational time, solution quality, robustness, stability, etc., requiring performance analysis and benchmarking tools.</li>

<li>The solutions may need to be implemented and executed in real systems, requiring system design and integration issues such as user interface, data input/output, communication protocol, feedback mechanism, etc.</li>

</ul>

<p>Some of the general purpose procedures for deterministic and stochastic scheduling are:</p>

<ul>

<li>Mathematical programming: This is a method that formulates the scheduling problem as an optimization problem using variables, constraints, and objective functions. The optimization problem can be solved using various techniques such as linear programming (LP), integer programming (IP), mixed-integer programming (MIP), nonlinear programming (NLP), etc.</li>

<li>Heuristics: These are methods that use simple rules or strategies to generate feasible or near-optimal solutions quickly without guaranteeing optimality or quality. Some examples of heuristics are priority rules (such as SPT, EDD, WSPT), dispatching rules (such as first come first served (FCFS), shortest queue (SQ), longest idle time (LIT)), etc.</li>

<li>Metaheuristics: These are methods that use general frameworks or principles to guide the search for better solutions by exploring the solution space using various operators such as initialization, selection, crossover, mutation, local search, acceptance criterion, etc. Some examples of metaheuristics are genetic algorithms (GA), simulated annealing (SA), tabu search (TS), ant colony optimization (ACO), particle swarm optimization (PSO), etc.</li>

<li>Simulation: This is a method that mimics the behavior of a real system using a computer model that captures its essential features and dynamics. Simulation can be used to evaluate and compare different scheduling scenarios and policies under various conditions and uncertainties.</li>

</ul>

<h5>Conclusion</h5>

<p>In this ar