Job shop scheduling
Authors: Carly Cozzolino, Brianna Christiansen, Elaine Vallejos, Michael Sorce, Jeff Visk (SYSEN 5800 Fall 2021)
The Job-Shop Scheduling Problem (JSSP) is a combinatorial optimization problem, which mainly concerns the operations industry. The aim of the problem is to optimize the schedule for allocation of shared resources over time to competing activities.
Within a JSSP you will have n number of jobs (J1, J2, ..., Jn ), that will need to be completed using a m number shared resources, most commonly denoted as machines (M1, M2, ..., Mm ). Each job will have operations (O) that will need to be completed in a specific order in order for the job to be completed. Operations must be completed at specific machines and require a specific amount of processing time (p) on that machine.Machines are assumed to only be able to process one operation at a time, driving the need to optimize the order in which these operations are completed. The goal of the JSSP is to minimize the overall time it takes to complete all n jobs within the problem.
The job shop scheduling problem is NP-hard meaning it’s complexity class for non-deterministic polynomial-time is least as hard as the hardest of problems in NP. As an input, there is a finite set J of jobs and a finite set of M of machines. Some jobs may need to be processed through multiple machines giving us an optimal potential processing order for each job. The processing time of the job j on machine m can be defined as the nonnegative integer . In order to minimize total completion time, the job shop can find the optimal schedule of J on M.
The assumptions made for solving the problem are as follows. The number of operations for each job is finite. The processing time for each operation in a particular machine is defined. There is a pre-defined sequence of operations that has to be maintained to complete each job. Delivery times of the products are undefined. No setup cost and tardiness cost. A machine can process only one job at a time. Each job visits each machine only once. No machine can deal with more than one type of task. The system cannot be interrupted until each operation of each job is finished. No machine can halt a job and start another job, before finishing the previous one. Given these assumptions, the problem can be solved at complexity class NP-hard.
One way of visualizing the job shop scheduling problem is using a Gantt-Chart. Given a problem of size 3 x 3 (J x M), a solution can be represented as shown in Figure 1. Depending on the job, the operating time on each machine could vary.
The Job-Shop Scheduling problem has been studied and reviewed over the past 60 years, with the interest in advanced methods ever growing. This research caused a wide variety of methods to come to be. Despite this effort, many approaches require further investigation to develop them into more attractive methods for use. Due to the limitations of a single method technique when solving complex problems, techniques can be combined to create a more robust and computationally strong process. In addition to multi-method techniques, dynamic methods are being developed to help account for unforeseen production issues. There are two categories which define these dynamic methods: approximate methods and precise methods. The best results currently can be found using approaches known as meta-heuristics or iterated local search algorithms. This approach combines a meta strategy with methods used in myopic problems. With a plethora of methods, some of the more notable techniques include the Genetic Algorithms, Simulated Annealing, and Tabu Search which are all complementary as opposed to competitive.
Stepping back to one of the earliest recorded works in the history of scheduling theory, there is an algorithm that focuses on efficiency. It is known as the first example of an efficient method. It is the work of David Stifler Johnson in his findings as documented in The Complexity of Flow Shop and Job-Shop Scheduling, Mathematics of Operations Research article published in 1954 . Johnson was an American computer scientist who specialized in optimization algorithms. The Flow Shop Method exemplifies efficiency in that it possesses a set of rules or requirement which increases polynomially with the input size to produce an optimized output. This method was designed to minimize the flow time of two machines. Johnson's work later played a significant role in subsequent Job Shop Scheduling problems due to the criterion to minimize the makespan. One of the beneficiaries of Johnson's work was Sheldon B. Akers in his article entitled A Graphical Approach to Production Scheduling Problems in 1956. In this article, he discusses his method which uses a conventional XY coordinate system to plot 2 parts of a problem with each part being an axis. By shading in the rectangle which relates to the operations on each axis and drawing a line to point P to create the program line. After analyzing the graph, one can conclude that the shortest line corresponds to the program which completes both parts in the least amount of time.
Despite the high number of potential variables involved in the Job-Shop Problem, one of the more commonly known methods which strived for efficiency was used in solving uniform length jobs with 2 processor systems. This competitive method published in 1972 is called the Coffman-Graham algorithm, and is explained in their article Optimal scheduling for two-processor systems. Named after Edward G. Coffman, Jr. and Ronald Graham, the problem makes a sequence of levels by organizing each element of a partially ordered set. This algorithm assigns a latter element a lower sequence or level, which ensures that each individual level has a distinct and fixed bound width. When that width is equal to 2, the number of levels is at a minimum.
One common method is a technique that can be applied to other MILP problems outside of the Job-Shop Scheduling problem, called the Branch and Bound method. The Branch and Bound method can be used to solve simpler problems such as those with only one machine. This method is used for integer programing problems and is an enumerative technique.  An example of this is detailed below. For more complicated problems, a dynamic solution is required, one of which combines the Branch and Bound method with the Cutting-Plane method. The Branch and Bound generally presents a problem tree in which the branches are traversed using the lower bound calculated to find the optimal solution. The problem is recursively divided into sub-problems as necessary. Prior to enumerating, the feasible solutions are compared to find the smallest bound. The lowest feasible solution is then bounded upon, adding the constraint to restrict the set of possible solutions. These steps are bounded upon until the final result has yielded the optimized solution.
Job-Shop Scheduling - CNC Machine Numerical Example
This example demonstrates how a Job-Shop Scheduling problem can be solved using the Branch and Bound method explained above. 
There are three mechanical parts that need to be processed (jobs) on a single available CNC machine. Each job has a finite duration to complete and a due date in the overall project schedule. This information is shown in the following table:
|Job||Duration (days)||Due Date|
The goal of this problem is to determine which sequence the jobs should be completed in to minimize total accrued project delay. The following table shows an example to calculate delay for one particular sequence, ABC.
|A||6||d1 = 6-8 = -2 -> 0|
|A, B||6+4 = 10||d2 = 10- 4 = 6|
|A, B, C||6+4+5||d3 = 15 - 12 = 3|
By beginning with job A at day 0, it will be completed by day 6. This is 2 days before the job A deadline and adds no delay to our overall schedule. At completion of job B we add the 6 days used on job A to the 4 needed for job B to get a completion time of 10 days. This yields a 6 day delay, as the deadline for job B was day 4. Finally we repeat that process for job C and see an additional 3 day delay. The total project delay T is equal to the summation of the accrued delay for each task: T = d1 + d2 + d3 = 0 + 6 + 3 = 9 days.
Using the Branch and Bound method, we can expand upon this concept and determine the sequence yielding the smallest delay out of all feasible solutions. We first define our decision variables where Xij = 1 if job j is put in the ith position, and Xij = 0 otherwise. Positions of jobs in the sequence are denoted by i = 1, 2, 3; Jobs are denoted by j = A, B, C.
The following explanation can be visualized in the Branch and Bound flow diagram. Regardless of the sequence, we know that the last job will always be completed on day 15. With this information, we can calculate which job will result in the smallest possible delay if put third in the sequence. For our example, that is task C, showing that the smallest the delay can be is 3 days. From there we branch again at the node with the smallest possible delay and repeat the process for determining the best choice for the second job in the sequence. Here we see that Job A in the second position yields a delay of at least 5. Finally, with only one option available for the job in the first position, we have reached our optimal sequence: BAC. This sequence minimized delay to 5 days, and thus is the optimal solution.
There are several ways in which the job-shop scheduling problem can be modified, often to simplify the problem at hand, for a variety of applications. However, there are many distinctions that can be made between every utilization of the JSSP so there is continuous research ongoing to improve adaptability of the problem.
The most obvious example can be found in the manufacturing industry, as the name suggests. It is common for some production jobs to require certain machines to perform tasks, due to the proper capabilities or equipment of a given machine. This adds an additional layer of complexity to the problem, because not any job can be processed on any machine. This is known as flexible manufacturing. Furthermore, there are many uncertain factors that are not accounted for in the basic understanding of the problem, such as delays in delivery of necessary supplies, significant absence of workers, equipment malfunction, etc. 
This problem can also be applied to many projects in the technology industry. In computer programming, it is typical that computer instructions can only be executed one at a time on a single processor, sequentially. In this example of multiprocessor task scheduling, the instructions can be thought of as the "jobs" to be performed and the processors required for each task can be compared to the "machines". Here we would want to schedule the order of instructions such that the number of operations performed is maximized to make the computer as efficient as possible. Within this application, there are a further variety of heuristic methods considered as to how to execute tasks to maximize efficiency and minimize costs, including minimum completion time, duplex, max-min, min-min, tabu search, and genetic algorithm. Each of these methods have their advantages and disadvantages, weighing the importance of minimizing execution time versus finding the closest to optimal solution.
With the progression of automation in recent years, robotic tasks such as moving objects from one location to another are similarly optimized. In this application, the extent of movement of the robot is minimized while conducting the most amount of transport jobs to support the most effective productivity of the robot. There are additional nuances within this problem as well, such as the rotational movement capabilities of the robot, the layout of the robots within the system, and the number of parts or units the robot is able to produce. While each of these factors contribute to the complexity of the job shop scheduling problem, they are all worth considering as more and more companies are proceeding in this direction to reduce labor costs and increase flexibility and safety of the enterprise.
Job-Shop Scheduling Problems are NP-hard combinatorial optimization problems, predominately used for operations applications. JSSPs minimize the overall time a set of jobs takes to complete, by optimizing the schedule in which shared resources are set to be allocated.
JSSPs can be solved using a variety of methods and algorithms.
There are multiple applications for JSSPs, the predominate application is within the manufacturing industry as the name implies. Other applications for this problem could also be the optimization of a computer's processing power as it executes multiple programs, optimization of automated equipment or robots, as well as a number of other applications.
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