There are many possible objectives in constructing a schedule, including

Job shop scheduling is also known as shop floor control (SFC), production control, and production activity control (PAC). Regardless of their primary scheduling objective, manufacturers typically have a production control department whose responsibilities consist of

Managers have multiple, conflicting scheduling objectives.

• Shop floor control (SFC): the scheduling and monitoring of day-to-day production in a job shop. It is usually performed by the production control department.

• Load leveling: the process of smoothing out the work assigned.

• Dispatch list: a shop paper that specifies the sequence in which jobs should be processed.

Loading is the process of assigning work to limited resources. Many times an operation can be performed by various persons, machines, or work centers but with varying efficiencies. If there is enough capacity, each worker should be assigned to the task that he or she performs best, and each job to the machine that can process it most efficiently. In effect, that is what happens when CRP generates a load profile for each machine center. The routing file used by CRP lists the ma chine that can perform the job most efficiently first. If no overloads appear in the load profile, then production control can proceed to the next task of sequencing the work at each center. However, when resource constraints produce overloads in the load profile, production control must examine the list of jobs initially assigned and decide which jobs to reassign elsewhere. The problem of determining how best to allocate jobs to machines or workers to tasks can be solved with the assignment method of linear programming.

• Loading: the process of assigning work to limited resources.

THE ASSIGNMENT METHOD OF LOADING

The assignment method is a specialized linear programming solution procedure for deciding which worker to assign to a task, or which job to assign to a machine. (See Supplement 14 for linear programming.) Given a table of tasks and resources, the procedure creates an opportunity cost matrix and selects the best assignment in consideration of tradeoffs among alternatives. With this technique, only one job may be assigned to each worker or machine. The procedure for a minimization problem is outlined as follows:

The assignment method of loading is a form of linear programming.

1. Perform row reductions by subtracting the minimum value in each row from all other row values.

2. Perform column reductions by subtracting the minimum value in each column from all other column values.

3. The resulting table is an opportunity cost matrix. Cross out all zeros in the matrix using the minimum number of horizontal or vertical lines.

4. If the number of lines equals the number of rows in the matrix, an optimal solution has been reached and assignments can be made where the zeros appear. Otherwise, modify the matrix by subtracting the minimum uncrossed value from all other uncrossed values and adding this same amount to all cells where two lines intersect. All other values in the matrix remain unchanged.

5. Repeat steps 3 and 4 until an optimal solution is reached.

Assigning Work at WebStar

WebStar, Inc. has four Web projects to complete and four workers with varying degrees of expertise in Web development for particular industries. Estimates of processing times (in hours) for each project by each worker are shown below. Development time costs an average of $100 an hour. Assign each worker to a project so that cost is minimized.

Initial Matrix	Project 1	Project 2	Project 3	Project 4
Bryan	10	5	6	10
Kari	6	2	4	6
Noah	7	6	5	6
Chris	9	5	4	10

Solution

1. Row reduction—Find the best assignment in each row. Subtract the smallest value in each row from all other row values. The resulting number is the opportunity cost of assigning a worker to that project. For example, the best assignment for Bryan would be project 2. Thus, its value in the following matrix is zero. However, if Bryan were assigned to project 1 or 4, it would take (10 – 5) = 5 hours longer to complete, and if assigned to project 3, one hour longer.

   Row Reduction	Project 1	Project 2	Project 3	Project 4
   Bryan	5	0	1	5
   Kari	4	0	2	4
   Noah	2	1	0	1
   Chris	5	1	0	6

2. Column reduction—Find the best assignment in each column. Subtract the smallest value in each column from all other values in the column. For example, project 1 can be completed the fastest with Noah as project leader, so it has an opportunity cost of zero. Assigning Bryan to project 1 would require 3 more hours of processing; thus, its opportunity cost is 3.

   Column Reduction	Project 1	Project 2	Project 3	Project 4
   Bryan	3	0	1	4
   Kari	2	0	2	3
   Noah	0	1	0	0
   Chris	3	1	0	5

3. Look for unique assignments—Examine the matrix and cover all zeroes. Remember that each person can only be assigned to one project and each project can only have one leader. A problem occurs when project 2 is the best for both Bryan and Kari, and project 3 is best for both Noah and Chris. In the following matrix, rows or columns containing optimal assignments are highlighted. This is called "covering all zeroes." If the number of lines (or highlighting) is less than the number of rows, the matrix needs to be modified.

   

4. Modify the matrix—Find the smallest value of the entries that are not highlighted. Subtract that value from all other non-highlighted entries, and add it to the entries where the lines intersect. Below we have subtracted 2 from the noncovered entries and added it to the intersection points (i.e., Noah's project 2 and project 3 values).

   

5. Look for unique assignments—Assign each worker to the best project available. Bryan is assigned to project 2. With project 2 eliminated, Kari is assigned to project 1. Since project 1 has been eliminated, Noah is assigned to project 4. That leaves Chris for project 3.

   

6. Calculate performance—Referring back to the original assignment matrix, the times required to finish each project are given below. At a cost of $100 hour, the projects will cost (5 + 6 + 4 + 6) × $100 = $2100 to complete.

   

Figure E17.1. The Assignment Method

Smaller assignment problems, such as our example, are usually solved by hand. Larger ones can be solved in Excel with Solver, as shown in Exhibit 17.1. Assignment problems may also involve maximizing profit or customer satisfaction. When solving maximization problems by hand, each entry in the initial matrix should be subtracted from the largest matrix value before proceeding as a minimization problem. When solving with Excel, simply change the Solver objective function from min to max.

When more than one job is assigned to a machine or activity, the operator needs to know the order in which to process the jobs. The process of prioritizing jobs is called sequencing. If no particular order is specified, the operator would probably process the job that arrived first. This default sequence is called first-come, first-served (FCFS). If jobs are stacked on arrival to a machine, it might be easier to process the job first that arrived last and is now on top of the stack. This is called last-come, first-served (LCFS) sequencing.

• Sequencing: prioritizes jobs that have been assigned to a resource.

Another common approach is to process the job first that is due the soonest or the job that has the highest customer priority. These are known as earliest due date (DDATE) and highest customer priority (CUSTPR) sequencing. Operators may also look through a stack of jobs to find one with a similar setup to the job that is currently being processed (SETUP). That would minimize the down-time of the machine and make the operator's job easier.

Variations on the DDATE rule include minimum slack (SLACK) and smallest critical ratio (CR). SLACK considers the work remaining to be performed on a job as well as the time remaining (until the due date) to perform that work. Jobs are processed first that have the least difference (or slack) between the two, as follows:

A sampling of heuristic sequencing rules.

The critical ratio uses the same information as SLACK, but recalculates the sequence as processing continues and arranges the information in ratio form. Mathematically, the CR is calculated as follows:

If the work remaining is greater than the time remaining, the critical ratio will be less than 1. If the time remaining is greater than the work remaining, the critical ratio will be greater than 1. If the time remaining equals work remaining, the critical ratio exactly equals 1. The critical ratio allows us to make the following statements about our schedule:

Other sequencing rules examine processing time at a particular operation and order the work either by shortest processing time (SPT) or longest processing time (LPT). LPT assumes long jobs are important jobs and is analogous to the strategy of doing larger tasks first to get them out of the way. SPT focuses instead on shorter jobs and is able to complete many more jobs earlier than LPT. With either rule, some jobs may be inordinately late because they are always put at the back of a queue.

All these "rules" for arranging jobs in a certain order for processing seem reasonable. We might wonder which methods are best or if it really matters which jobs are processed first anyway. Perhaps a few examples will help answer those questions.

SEQUENCING JOBS THROUGH ONE PROCESS

The simplest sequencing problem consists of a queue of jobs at one machine or process. No new jobs arrive to the machine during the analysis, processing times and due dates are fixed, and setup time is considered negligible. For this scenario, the completion time (also called flow time) of each job will differ depending on its place in the sequence, but the overall completion time for the set of jobs (called the makespan) will not change. Tardiness is the difference between a late job's due date and its completion time. Even in this simple case, there is no sequencing rule that optimizes both processing efficiency and due date performance. Let's consider an example.

• Flow time: the time it takes a job to flow through the system.

• Makespan: the time it takes for a group of jobs to be completed.

• Tardiness: the difference between the late job's due date and its completion time.

Simple Sequencing Rules

Because of the approaching holiday season, Joe Palotty is scheduled to work seven days a week for the next two months. October's work for Joe consists of five jobs, A, B, C, D, and E. Job A takes five days to complete and is due on day 10, job B takes ten days to complete and is due on day 15, job C takes two days to process and is due on day 5, job D takes eight days to process and is due on day 12, and job E, which takes six days to process, is due on day 8.

There are 120 possible sequences for the five jobs. Clearly, enumeration is impossible. Let's try some simple sequencing rules. Sequence the jobs by (a) first-come, first-served (FCFS), (b) earliest due date (DDATE), (c) minimum slack (SLACK), and (d) shortest processing time (SPT). Determine the completion time and tardiness of each job under each sequencing rule. Should Joe process his work as is—first-come, first-served? If not, what sequencing rule would you recommend to Joe?

Solution

Prepare a table for each sequencing rule. Start the first job at time 0. (When today's date is not given, assume it is day 0.) Completion time is the sum of the start time and the processing time. The start time of the next job is the completion time of the previous job.

1. FCFS: Process the jobs in order of their arrival, A, B, C, D, E.

   

2. DDATE: Sequence the jobs by earliest due date.

   

3. SLACK: Sequence the jobs by minimum slack. The slack for each job is calculated as: (due date – today's date) – remaining processing time.

   Job A (10 – 0) – 5 = 5*

   B (15 – 0) – 10 = 5*

   C (5 – 0) – 2 = 3

   D (12 – 0) – 8 = 4

   E (8 – 0) – 6 = 2

   *break the tie arbitrarily

   

4. SPT: Sequence the jobs by smallest processing time.

Summary

All the sequencing rules complete the month's work by day 31, as planned. However, no sequencing rule is able to complete all jobs on time. The performance of FCFS is either met or exceeded by DDATE and SPT. Thus, Joe should take the time to sequence this month's work.

Whether Joe sequences his work by DDATE or SPT depends on the objectives of the company for whom he works. The particular jobs that are tardy may also make a difference.

The Excel solution to this problem is shown in Exhibit 17.2.

Are the preceding results a function of this particular example, or are they indicative of the types of results we will get whenever these rules are applied? Analytically, we can prove that for a set number of jobs to be processed on one machine, the SPT sequencing rule will minimize mean job completion time (also known as flowtime) and minimize mean number of jobs in the system. On the other hand, the DDATE sequencing rule will minimize mean tardiness. No definitive statements can be made concerning the performance of the other sequencing rules.

There is no one sequencing rule that optimizes both processing efficiency and due date performance.

SEQUENCING JOBS THROUGH TWO SERIAL PROCESSES

Since few factories consist of just one process, we might wonder if techniques exist that will produce an optimal sequence for any number of jobs processed through more than one machine or process. Johnson's rule finds the fastest way to process a series of jobs through a two-step system in which every job follows the same sequence through two processes. Based on a variation of the SPT rule, it requires that the sequence be "mapped out" to determine the final completion time, or makespan, for the set of jobs. The procedure is as follows:

• Johnson's rule: gives an optimal sequence for jobs processed serially through two processes.

1. List the time required to complete each job at each process. Set up a one-dimensional matrix to represent the desired sequence with the number of slots equal to the number of jobs.

2. Select the smallest processing time at either process. If that time occurs at process 1, put the associated job as near to the beginning of the sequence as possible.

   

   Figure E17.2. Using Excel For Sequencing Rules

   

3. If the smallest time occurs at process 2, put the associated job as near to the end of the sequence as possible.

4. Remove the job from the list.

5. Repeat steps 2–4 until all slots in the matrix have been filled or all jobs have been sequenced.

Johnson's Rule

Johnson's Fine Restorations has received a rush order to refinish five carousel animals—an alligator, a bear, a cat, a deer, and an elephant. The restoration involves two major processes: sanding and painting. Mr. Johnson takes care of the sanding; his son does the painting. The time required for each refinishing job differs by the state of disrepair and degree of detail of each animal. Given the following processing times (in hours), determine the order in which the jobs should be processed so that the rush order can be completed as soon as possible.

Job	Process 1	Process 2
A	6	8
B	11	6
C	7	3
D	9	7
E	5	10

Solution

The smallest processing time, three hours, occurs at process 2 for job C, so we place job C as near to the end of the sequence as possible. C is now eliminated from the job list.

The next smallest time is five hours. It occurs at process 1 for job E, so we place job E as near to the beginning of the sequence as possible. Job E is eliminated from the job list.

The next smallest time is six hours. It occurs at process 1 for job A and at process 2 for job B. Thus, we place job A as near to the beginning of the sequence as possible and job B as near to the end of the sequence as possible. Jobs A and B are eliminated from the job list.

The only job remaining is job D. It is placed in the only available slot, in the middle of the sequence.

This sequence will complete these jobs faster than any other sequence. The following bar charts (called Gantt charts) are used to determine the makespan or final completion time for the set of five jobs. Notice that the sequence of jobs (E, A, D, B, C) is the same for both processes and that a job cannot begin at process 2 until it has been completed at process 1. Also, a job cannot begin at process 2 if another job is currently in process. Time periods during which a job is being processed are labeled with the job's letter. The gray shaded areas represent idle time.

The completion time for the set of five jobs is 41 hours. Note that although Johnson's rule minimizes makespan and idle time, it does not consider job due dates in constructing a sequence, so there is no attempt to minimize job tardiness.

The Excel solution to this problem is shown in Exhibit 17.3.

Figure E17.3. Using Excel for Johnson's Rule

In a job shop environment, where jobs follow different paths through the shop, visit many different machine centers, and compete for similar resources, it is not always easy to keep track of the status of a job. When jobs are first released to the shop, it is relatively easy to observe the queue that they join and predict when their initial operations might be completed. As the job progresses, however, or the shop becomes more congested, it becomes increasingly difficult to follow the job through the system. Competition for resources (resulting in long queues), machine breakdowns, quality problems, and setup requirements are just a few of the things that can delay a job's progress.

Shop paperwork, sometimes called a work package, travels with a job to specify what work needs to be done at a particular work center and where the item should be routed next. Workers are usually required to sign off on a job, indicating the work they have performed either manually on the work package or electronically through a PC located on the shop floor. Bar code technology and RFID tags have made this task easier by eliminating much of the tedium and errors of entering the information by computer keyboard. In its simplest form, the bar code is attached to the work package, which the worker reads with a wand at the beginning and end of his or her work on the job. In other cases, an RFID tag is attached to the pallet or crate that carries the items from work center to work center. The tag is read automatically as it enters and leaves the work area. The time a worker spends on each job, the results of quality checks or inspections, and the utilization of resources can also be recorded in a similar fashion.

For the information gathered at each work center to be valuable, it must be up-to-date, accurate, and accessible to operations personnel. The monitoring function performed by production control takes this information and transforms it into various reports for workers and managers to use. Progress reports can be generated to show the status of individual jobs, the availability or utilization of certain resources, and the performance of individual workers or work centers. Exception reports may be generated to highlight deficiencies in certain areas, such as scrap, rework, shortages, anticipated delays, and unfilled orders. Hot lists show which jobs receive the highest priority and must be done immediately. A well-run facility will produce fewer exception reports and more progress reports. In the next two sections we describe two such progress reports, the Gantt chart and the input/output control chart.

• Work package: shop paperwork that travels with a job.

GANTT CHARTS

Gantt charts, used to plan or map out work activities, can also be used to monitor a job's progress against the plan. As shown in Figure 17.1, Gantt charts can display both planned and completed activities against a time scale. In this figure, the dashed line indicating today's date crosses over the schedules for job 12A, job 23C, and job 32B.

From the chart we can quickly see that job 12A is exactly on schedule because the bar monitoring its completion exactly meets the line for the current date. Job 23C is ahead of schedule and job 32B is behind schedule.

• Gantt charts: show both planned and completed activities against a time scale.

Figure 17.1. A Gantt Chart

Gantt charts have been used since the early 1900s and are still popular today. They may be created and maintained by computer or by hand. In some facilities, Gantt charts consist of large scheduling boards (the size of several bulletin boards) with magnetic strips, pegs, or string of different colors that mark job schedules and job progress for the benefit of an entire plant. Gantt charts are a common feature of project management software, such as Microsoft Project.

INPUT/OUTPUT CONTROL

Input/output (I/O) control monitors the input to and output from each work center. Prior to such analysis, it was common to examine only the output from a work center and to compare the actual output with the output planned in the shop schedule. Using that approach in a job shop environment in which the performance of different work centers is interrelated may result in erroneous conclusions about the source of a problem. Reduced output at one point in the production process may be caused by problems at the current work center, but it may also be caused by problems at previous work centers that feed the current work center. Thus, to identify more clearly the source of a problem, the input to a work center must be compared with the planned input, and the output must be compared with the planned output. Deviations between planned and actual values are calculated, and their cumulative effects are observed. The resulting backlog or waiting line of work to be completed is monitored to ensure that it stays within a manageable range.

• Input/output (I/O) control: monitors the input and output from each work center.

Gantt charts have been used for more than 75 years to plan and monitor schedules. Today Gantt charts are more widely used than ever, often as part of the action plan from a quality-improvement team. In some factories, Gantt charts appear on large magnetic boards, displaying the plant's see. Computerized versions chart time, resources, and precedence requirements in an easy-to-read visual format. This Gantt chart is for publishing a textbook.

The input rate to a work center can be controlled only for the initial operations of a job. These first work centers are often called gateway work centers, because the majority of jobs must pass through them before subsequent operations are performed. Input to later operations, performed at downstream work centers, is difficult to control because it is a function of how well the rest of the shop is operating—that is, where queues are forming and how smoothly jobs are progressing through the system. The deviation of planned to actual input for downstream work centers can be minimized by controlling the output rates of feeding work centers. The use of input/output reports can best be illustrated with an example.

Input/Out Control

Hall Industries has begun input/output planning for its work centers. Below are the planned inputs and outputs for Work Center 5.

1. If production proceeds as planned, what will be the backlog at the end of period 4?

2. If actual input values are 60, 60, 65, 65 for periods 1 through 4, respectively, and output values cannot exceed 75, how much output can be expected from Work Center 5?

3. Is there a problem with production at Work Center 5?

Input/Output Report for Work Center 5

Solution

1. 

   If everything goes as planned, the backlog will be zero by period 4.

2. 

   With the reduced input, the backlog is worked off sooner, but the total production cannot keep pace with what was planned.

3. Although Work Center 5 has produced only 280 units instead of the planned 300, the problem appears to be with the process that feeds Work Center 5. Notice the deviations from planned are the same for both the input and output values. An Excel solution to this example is shown in Exhibit 17.4.

Figure E17.4. Using Excel for Input/Output Control

Input/output control provides the information necessary to regulate the flow of work to and from a network of work centers. Increasing the capacity of a work center that is processing all the work available to it will not increase output. The source of the problem needs to be identified. Excessive queues, or backlogs, are one indication that bottlenecks exist. To alleviate bottleneck work centers, the problem causing the backlog can be worked on, the capacity of the work center can be adjusted, or input to the work center can be reduced. Increasing the input to a bottleneck work center will not increase the center's output. It will merely clog the system further and create longer queues of work-in-process.

MARGIE DECK Plant Manager

I began working for a manufacturing company as a third-shift supervisor 15 years ago, straight out of grad school. I had no desire to work in a factory and just took the job to gain what I thought would be useful experience for pursuing a career in Human Resources. But I loved it from the start—manufacturing is fast-paced and exciting. There are deadlines to meet and real work to be done. No sitting behind a computer all day writing reports. Nothing is routine. We are always trying to get work done better or faster or cheaper than before. Our customers are demanding, and so are our stockholders.

We make engine bearings at this plant—lots of them, 88 million bearings a year—for a total of six customers in the automotive and industrial machinery industry. We use so many of the things you learn in an operations class—scheduling, lean production, theory of constraints, and tons of quality tools, like Pareto charts, root cause analysis, process improvement teams, and SPC. Quality and safety are our biggest concerns. sor talk about Six Sigma quality. Well, our defect rate is even better, at 2 parts per million. But if even one bad part goes to a customer's plant, it could cause a line shutdown and we'd be charged $500 a minute until the line is up and running again!

We have a very close relationship with our customers. They each have their own personality and ways of doing business. These are longstanding customers. If we get the bid for an engine program, we have it for the life of the product. The contract might say that we should be capable of producing 100,000 bearings per day. We'll be given a three-month forecast of usage that we use for manpower planning, and a two-week forecast for material planning. But day-to-day usage can vary significantly. Every morning at 2 a.m. our customers send us their orders for the day and tell us when a truck will be arriving to pick them up. We check the EDI system when we get in and start to work putting together the day's schedule. There's always a challenqe, always something new.

The process for scheduling that we have described thus far in this chapter, loading work into work centers, leveling the load, sequencing the work, and monitoring its progress, is called infinite scheduling. The term infinite is used because the initial loading process assumes infinite capacity. Leveling and sequencing decisions are made after overloads or underloads have been identified. This iterative process is time-consuming and not very efficient.

An alternative approach to scheduling called finite scheduling assumes a fixed maximum capacity and will not load the resource beyond its capacity. Loading and sequencing decisions are made at the same time, so that the first jobs loaded onto a work center are of highest priority. Any jobs remaining after the capacity of the work center or resource has been reached are of lower priority and are scheduled for later time periods. This approach is easier than the infinite scheduling approach, but it will be successful only if the criteria for choosing the work to be performed, as well as capacity limitations, can be expressed accurately and concisely.

• Infinite scheduling: loads without regard to capacity, then levels the load and sequences the jobs.

• Finite scheduling: sequences jobs as part of the loading decision. Resources are never loaded beyond capacity.

Finite scheduling systems use a variety of methods to develop their schedules, including mathematical programming, network analysis, simulation, constraint-based programming, genetic algorithms, neural networks, and expert systems. Because the scheduling system, not the human scheduler, makes most of the decisions, considerable time is spent incorporating the special characteristics and requirements of the production system into the database and knowledge base of the scheduling software. While some companies will develop their own finite scheduling software, most will purchase generic or industry-specific scheduling software as an add-on to their ERP system. This class of scheduling software, with libraries of algorithms and heuristics from which to choose, has become known as advanced planning and scheduling (APS). SAP's APS system is called the Advanced Planner and Optimizer (APO), and i2 Technologies' is called Factory Planner. These systems also support collaborative planning and scheduling with trading partners. Both APO and Factory Planner use constraint-based programming and genetic algorithms to develop schedules. Figure 17.2 applies these techniques to sequence a set of jobs so that setup time is minimized.

• Advanced planning and scheduling (APS): a software system that uses intelligent analytical tools and techniques to develop realistic schedules.

Constraint-based programming (CBP) identifies a solution space and then systematically evaluates possible solutions subject to constraints on the system. In scheduling, when one job or person is assigned, the options for additional job assignments are reduced or further constrained. This cascading effect on the remaining jobs to be scheduled is called constraint propagation. In part (a) of Figure 17.2, all possible sequences are evaluated, with A-C-B and C-A-B yielding the best solutions.

Figure 17.2. Advanced Planning and Scheduling Techniques: Source: SAP AG, Advanced Planner and Optimizer, company brochure, 1999.

Genetic algorithms are based on the natural selection properties of genetics. Starting from a feasible solution, alternative solutions or "offspring" are generated with slightly different characteristics. These are evaluated against an objective function and compared to subsequent solutions. The number of generations and number of offspring per generation are specified by the user. In this example, two generations are produced with two offspring per generation. That's more than enough offspring to try all possible combinations. Again, A-C-B and C-A-B are identified as optimal sequences.

As schedules are executed, another type of software, aptly named a manufacturing execution system (MES), monitors work status, machine status, material usage and availability, and quality data. Alerts are sent to the APS system that shop conditions have changed and a scheduling revision is required.

Scheduling problems can be enormous in size, especially as the number of product options proliferate and linked schedules along a supply chain are considered. Some relief has come from the availability of more powerful computers and the use of artificial intelligence techniques. But the biggest impact has come from altered views of the production system. By grouping parts or products into families and scheduling bottleneck resources first, the scheduling problem is sufficiently reduced so that sophisticated solutions are feasible. In the next section, we present a common precept of today's scheduling systems, the theory of constraints.

• Genetic algorithm: method that generates possible solutions based on genetic combinations of previous solutions.

• manufacturing execution system (MES): manufacturing software that monitors operations, collects data, and controls processes on the shop floor.

In the 1970s, an Israeli physicist named Eliyahu Goldratt responded to a friend's request for help in scheduling his chicken coop business. Lacking a background in manufacturing or production theory, Dr. Goldratt took a commonsense, intuitive approach to the scheduling problem. He developed a software system that used mathematical programming and simulation to create a schedule that realistically considered the constraints of the manufacturing system. The software produced good schedules quickly and was marketed in the early 1980s in the United States. After more than 100 firms had successfully used the scheduling system (called OPT), the creator sold the rights to the software and began marketing the theory behind the software instead. He called his approach to scheduling the theory of constraints (TOC). General Motors and other manufacturers call its application synchronous manufacturing.

• Theory of constraints (TOC): a finite scheduling approach that concentrates on scheduling the bottleneck resource.

Decision making in manufacturing is often difficult because of the size and complexity of the problems faced. Dr. Goldratt's first insight into the scheduling problem led him to simplify the number of variables considered. He learned early that manufacturing resources typically are not used evenly. Instead of trying to balance the capacity of the manufacturing system, he decided that most systems are inherently unbalanced and that he would try to balance the flow of work through the system instead. He identified resources as bottleneck or nonbottleneck and observed that the flow through the system is controlled by the bottleneck resource. This resource should always have material to work on, should spend as little time as possible on nonproductive activities (e.g., setups, waiting for work), should be fully staffed, and should be the focus of improvement or automation efforts. Goldratt pointed out that an hour's worth of production lost at a bottleneck reduces the output of the system by the same amount of time, whereas an hour lost at a nonbottleneck may have no effect on system output.

From this realization, Goldratt was able to simplify the scheduling problem significantly. He concentrated initially on scheduling production at the bottleneck resource and then scheduling the nonbottleneck resources to support the bottleneck activities. Thus, production is synchronized, or "in sync," with the needs of the bottleneck and the system as a whole.

PROCESS VS. TRANSFER BATCH SIZES

Goldratt's second insight into manufacturing concerned the concept of lot sizes or batch sizes. Goldratt saw no reason for fixed lot sizes. He differentiated between the quantity in which items are produced, called the process batch, and the quantity in which the items are transported, called the transfer batch. Ideally, items should be transferred in lot sizes of one. The process batch size for bottlenecks should be large, to eliminate the need for setups. The process batch size for non-bottlenecks can be small because time spent in setups for nonbottlenecks does not affect the rest of the system.

The TOC scheduling procedure, illustrated in Example 17.5, follows these steps:

Process batch sizes and transfer batch sizes do not have to match.

1. Identify the bottleneck.

2. Schedule the job first whose lead time to the bottleneck is less than or equal to the bottleneck processing time.

3. Forward schedule the bottleneck machine.

4. Backward schedule the other machines to sustain the bottleneck schedule.

5. Transfer in batch sizes smaller than the process batch size.

Synchronous Manufacturing

The following diagram contains the product structure, routing, and processing time information for product A. The process flows from the bottom of the diagram upward. Assume one unit of items B, C, and D are needed to make each A. The manufacture of each item requires three operations at machine centers 1, 2, or 3. Each machine center contains only one machine. A machine setup time of 60 minutes occurs whenever a machine is switched from one operation to another (within the same item or between items).

Design a schedule of production for each machine center that will produce 100 A's as quickly as possible. Show the schedule on a Gantt chart of each machine center.

Solution

The bottleneck machine is identified by summing the processing times of all operations to be performed at a machine.

• Machine 2 is identified as the bottleneck, so we schedule machine 2 first. From the product structure diagram, we see that three operations are performed at machine 2—B2, C3, and D2. If we schedule item B first, B will reach machine 2 every five minutes (since B has to be processed through machine 1 first), but each B takes only three minutes to process at machine 2, so the bottleneck will be idle for two minutes of every five minutes. A similar result occurs if we schedule item D first on machine 2. The bottleneck will be idle for two minutes of every ten until D has finished processing. The best alternative is to schedule item C first. The first C will not reach machine 2 until time 12, but after that the bottleneck machine will remain busy.

  

• We begin our Gantt charts by processing item C through the three machine centers. As shown in Figure 17.3, the Gantt charts look different from our earlier examples because we allow each item to be transferred to the next operation immediately after it is completed at the current operation (i.e., the transfer batch size is 1). We will still process the items in batches of 100 to match our demand requirements. The gray shaded areas represent idle time between operations due to setup time requirements or because a feeding operation has not yet been completed.

  

  Figure 17.3. Gantt Chart Solution to Example 17.5

• C3 is completed at machine center 2 at time 1512. After setup, it is ready for a new item at time 1572. We have a choice between B2 and D2, since both B1 and D1 can be completed by 1572. Completion time at machine center 2 will be the same regardless of whether B2 or D2 is processed first; however, the completion time at the other machine centers (and thus for product A) will be affected by the bottleneck sequence. From the product structure diagram, we note that B3 can be completed more quickly than D3 because D3 must wait three minutes for D2 to be completed, whereas B3 will always have a queue of items from B2 to work on. Thus, we schedule B2 and then D2 on machine center 2.

• With the bottleneck sequence of C3, B2, D2 established, we can now schedule machine center 1 (C2, B1, B3) and machine center 3 (C1, D1, D3) in the same general order as the bottleneck sequence. The completion time for producing 100 A's is 2737 minutes. The total idle time at the three machine centers is 994 minutes.

Identify the bottleneck.

Process in batches of 100; transfer one-at-a-time.

Sustain the bottleneck.

Sequence the other machines to support the bottleneck sequence.

Labor is one of the most flexible resources. Workers can be hired and fired more easily than equipment can be purchased or sold. Labor-limited systems can expand capacity through overtime, expanded workweeks, extra shifts, or part-time workers. This flexibility is valuable but it tends to make scheduling difficult. Service firms especially spend an inordinate amount of time developing employee schedules. A supervisor might spend an entire week making up the next month's employee schedule. The task becomes even more daunting for facilities that operate on a 24-hour basis with multiple shifts.

Employee scheduling has lots of options because labor is a very flexible resource.

The assignment method of linear programming discussed earlier in this chapter can be used to assign workers with different performance ratings to available jobs. Large-scale linear programming is currently used by McDonald's to schedule its large part-time workforce. American Airlines uses a combination of integer linear programming and expert systems for scheduling ticket agents to coincide with peak and slack demand periods and for the complicated scheduling of flight crews. Although mathematical programming certainly has found application in employee scheduling, most scheduling problems are solved by heuristics (i.e., rules of thumb) that develop a repeating pattern of work assignments. Often heuristics are imbedded in a decision support system to facilitate their use and increase their flexibility. One such heuristic^[33] used for scheduling full-time workers with two days off per week, is given next.

Employee Scheduling Heuristic:

Scheduling employees, especially part-time workers, can take considerable time and effort. Most employers develop weekly or monthly schedules by hand, trying to balance the needs of employees, supervisors, and customers. One way to avoid the headaches of employee scheduling and improve customer responsiveness is to automate. That's what the banking industry did with ATMs that operate 24 hours a day. Now maintenance activities must be scheduled to ensure that the facility is available to the customer as promised!

The heuristic just illustrated can be adapted to ensure that the two days off per week are consecutive. Other heuristics schedule workers two weeks at a time, with every other weekend off.

AUTOMATED SCHEDULING SYSTEMS

Scheduling large numbers of workers at numerous locations requires a computerized scheduling system. Sophisticated employee scheduling software is available as a stand-alone system or as part of an ERP package. For example, Workbrain, part of Infor's ERP system, provides:

• Staff scheduling assigns qualified workers to standardize shift patterns taking into account leave requests and scheduling conflicts. The solutions include social constraints such as labor laws for minors, overtime payment regulations, and holidays or religious holidays that may differ by global location.

• Schedule bidding puts certain shift positions or schedule assignments up for bid by workers; allows workers to post and trade schedules with others as long as coverage and skill criteria are met.

• Schedule optimization creates demand-driven forecasts of labor requirements and assigns workers to variable schedules (in some cases, as small as 15 minutes blocks of time) that change dynamically with demand. Uses mathematical programming and artificial intelligence techniques.

Scheduling techniques vary by type of production process. Scheduling in a job shop environment is difficult because jobs arrive at varying time intervals, require different resources and sequences of operations, and are due at different times. This lowest level of scheduling is referred to as shop floor control or production control. It involves assigning jobs to machines or workers (called loading), specifying the order in which operations are to be performed, and monitoring the work as it progresses. Techniques such as the assignment method are used for loading, various rules whose performance varies according to the scheduling objective are used for sequencing, and Gantt charts and input/output control charts are used for monitoring.

Realistic schedules must reflect capacity limitations. Infinite scheduling initially assumes infinite capacity and then manually "levels the load" of resources that have exceeded capacity. Finite scheduling loads jobs in priority order and delays those jobs for which current capacity is exceeded. The theory of constraints is a finite scheduling approach that schedules bottleneck resources first and then schedules other resources to support the bottleneck schedule. It also allows items to be transferred between resources in lot sizes that differ from the lot size in which the item is produced. Other advanced planning and scheduling techniques include mathematical programming, genetic algorithms, and simulation.

Employee scheduling is often difficult because of the variety of options available and the special requirements for individual workers. Scheduling heuristics are typically used to develop patterns of worker assignment. Automated workforce scheduling systems are becoming more commonplace.

Minimum Slack

SLACK = (due date – today's date) – (processing time)

Critical Ratio

advanced planning and scheduling (APS) a software system that uses intelligent analytical tools and techniques to develop realistic schedules.

dispatch list a shop paper that specifies the sequence in which jobs should be processed; it is often derived from specific sequencing rules.

drum-buffer-rope a concept in theory of constraints where the drum sets the pace of production, buffer is placed in front of the bottleneck, and rope communicates changes.

finite scheduling an approach to scheduling that loads jobs in priority order and delays those jobs for which current capacity is exceeded.

flow time the time that it takes for a job to "flow" through the system; that is, its completion time.

Gantt chart a bar chart that shows a job's progress graphically or compares actual against planned performance.

genetic algorithms a method that generates possible solutions based on genetic combinations of previous solutions.

infinite scheduling an approach to scheduling that initially assumes infinite capacity and then manually "levels the load" of resources that have exceeded capacity.

input/output (I/O) control a procedure for monitoring the input to and output from a work center to regulate the flow of work through a system.

Johnson's rule an algorithm for sequencing any number of jobs through two serial operations to minimize makespan.

load leveling the process of smoothing out the work assigned across time and the available resources.

loading the process of assigning work to individual workers or machines.

makespan the time that it takes for a group of jobs to be completed—that is, the completion time of the last job in a group.

manufacturing execution systems (MES) manufacturing software that monitors operations, collects data, and controls processes on the shop floor.

scheduling the determination of when labor, equipment, and facilities are needed to produce a product or provide a service.

sequencing the process of assigning priorities to jobs so that they are processed in a particular order.

shop floor control (SFC) scheduling and monitoring day-to-day production in a job shop; also known as production control or production activity control.

tardiness the difference between a job's due date and its completion time for jobs completed after their due date.

theory of constraints a finite scheduling approach that differentiates between bottleneck and nonbottleneck resources and between transfer batches and process batches.

work package shop paperwork that travels with a job to specify what work needs to be done at a particular machine center and where the item should be routed next.

1. ASSIGNMENT PROBLEM

Wilkerson Printing has four jobs waiting to be run this morning. Fortunately, they have four printing presses available. However, the presses are of different vintage and operate at different speeds. The approximate times (in minutes) required to process each job on each press are given next. Assign jobs to presses to minimize the press running times.

SOLUTION

Row reduction:

Column reduction:

Cover all zeroes:

The number of lines equals the number of rows, so this is the final solution. Make assignments:

Assign job A to press 4, job B to press 1, job C to press 3, and job D to press 2. Refer to the original matrix for actual processing times. The total machining time required is (10 + 40 + 35 + 45) = 130 minutes.

2. JOHNSON'S RULE

Clean and Shine Car Service has five cars waiting to be washed and waxed. The time required (in minutes) for each activity is given next. In what order should the cars be processed through the facility? When will the batch of cars be completed?

SOLUTION

Use Johnson's rule to sequence the cars. The lowest processing time is two minutes for waxing car 2. Since waxing is the second operation, we place car 2 as near to the end of the sequence as possible, in last place. The next-lowest time is three minutes for washing car 5. Since washing is the first operation, we place car 5 as near to the front of the sequence as possible, in first place. The next-lowest time is five minutes for washing car 1 and waxing car 3. Car 1 is scheduled in second place, and car 3 is put in next-to-last place (i.e., fourth). That leaves car 4 for third place.

The completion time for washing and waxing the five cars is 35 minutes. The washing facility is idle for two minutes at the end of the cycle. The waxing facility is idle for three minutes at the beginning of the cycle and four minutes during the cycle.

17-1. How do scheduling activities differ for projects, mass production, and process industries?

17-2. Why is scheduling a job shop so difficult?

17-3. What three functions are typically performed by a production control department?

17-4. Give examples of four types of operations (manufacturing or service) and suggest which scheduling objectives might be appropriate for each.

17-5. How can the success of a scheduling system be measured?

17-6. Describe the process of loading and load leveling. What quantitative techniques are available to help in this process?

17-7. What is the purpose of dispatch lists? How are they usually constructed?

17-8. When should the following sequencing rules be used? (a) SPT; (b) Johnson's rule; (c) DDATE; (d) FCFS.

17-9. Give examples of sequencing rules you use to prioritize work.

17-10. What information is provided by the critical ratio sequencing rule? How does it differ from SLACK?

17-11. How are work packages, hot lists, and exception reports used in a job shop?

17-12. What are Gantt charts, and why are they used so often?

17-13. Explain the concept behind input /output control. Describe how gateway work centers, downstream work centers, and backlogs affect shop performance.

17-14. Explain the difference between infinite and finite scheduling.

17-15. How does theory of constraints differ from traditional scheduling? How should bottleneck resources and nonbottleneck resources be scheduled? Why should transfer batches and process batches be treated differently?

17-16. Explain the drum-buffer-rope concept.

17-17. Discuss the similarities and differences between theory of constraints and lean production.

17-18. What are some typical issues involved in employee scheduling?

17-19. What quantitative techniques are available to help develop employee schedules? What quantitative techniques are available for advanced planning and scheduling systems?

17-20. Look for advanced planning and scheduling software on the Internet. Write a summary of the techniques presented.

17-1. At Valley Hospital, nurses beginning a new shift report to a central area to receive their primary patient assignments. Not every nurse is as efficient as another with particular kinds of patients. Given the following patient roster, care levels, and time estimates, assign nurses to patients to optimize efficiency. Also, determine how long it will take for the nurses to complete their routine tasks on this shift.

17-2. Valley Hospital (from Problem 17-1) wants to focus on customer perceptions of quality, so it has asked its patients to evaluate the nursing staff and indicate preferences for assignment. Reassign the nursing staff to obtain the highest customer approval rating possible (a perfect score is 100).

Compare the results with those from Problem 17-1. What is the average rating of the assignment? What other criteria could be used to assign nurses?

17-3. Fibrous Incorporated makes products from rough tree fibers. Its product line consists of five items processed through one of five machines. The machines are not identical, and some products are better suited to some machines. Given the following production time in minutes per unit, determine an optimal assignment of product to machine:

17-4. Sunshine House received a contract this year as a supplier of Girl Scout cookies. Sunshine currently has five production lines, each of which will be dedicated to a particular kind of cookie. The production lines differ by sophistication of machines, site, and experience of personnel. Given the following estimates of processing times (in hours), assign cookies to lines to minimize the sum of completion times:

17-5. Karina Nieto works for New Products Inc., and one of her many tasks is assigning new workers to departments. The company recently hired six new employees and would like each one to be assigned to a different department. The employees have completed a two-month training session in each of the six departments from which they received the evaluations shown below (higher numbers are better). Determine how the new employees should be assigned to departments so that overall performance is maximized.

17-6. Evan Schwartz has six jobs waiting to be processed through his machine. Processing time (in days) and due date information for each job are as follows:

Sequence the jobs by FCFS, SPT, SLACK, and DDATE. Calculate the mean flow time and mean tardiness of the six jobs under each sequencing rule. Which rule would you recommend?

17-7. College students always have a lot of work to do, but this semester, Katie Lawrence is overwhelmed. Following are the assignments she faces, the estimated completion times (in days), and due dates:

Help Katie prioritize her work so that she completes as many assignments on time as possible. Today is November 2. How would your sequence of assignments change if Katie were interested in minimizing the mean tardiness of her assignments?

17-8. Today is day 4 of the planning cycle. Sequence the following jobs by FCFS, SPT, SLACK, and DDATE. Calculate the mean flow time and mean tardiness for each sequencing rule. Which rule would you recommend?

17-9. Alice's Alterations has eight jobs to be completed and only one sewing machine (and sewing machine operator). Given the processing times and due dates as shown here, prioritize the jobs by SPT, DDATE, and SLACK. Today is day 5.

Calculate mean flow time, mean tardiness, maximum tardiness, and number of jobs tardy for each sequence. Which sequencing rule would you recommend? Why?

17-10. Jobs A, B, C, and D must be processed through the same machine center. Sequence the following jobs by (a) SPT and (b) SLACK. Calculate mean flow time, mean tardiness, and maximum tardiness. Which sequencing rule would you recommend? Why?

17-11. Sequence the following jobs by (a) SPT, (b) DDATE, and (c) SLACK. Calculate mean flow time, mean tardiness, and maximum tardiness. Which sequencing rule would you recommend? Why?

17-12. Claims received by Healthwise Insurance Company are entered into the database at one station, and sent to another station for review. The processing time (in minutes) required for each general type of claim is shown here. Currently, Bill Frazier has 10 claims to be reviewed. In what order should he process the claims so that the entire batch can get into the system as soon as possible? How long will it take to process the 10 claims completely?

17-13. Jobs processed through Percy's machine shop pass through three operations: milling, grinding, and turning. The hours required for each of these operations is as follows:

Sequence the job by (a) FCFS and (b) shortest processing time (SPT). Make a Gantt chart for each machine and each rule. Which sequencing rule would you recommend?

17-14. Restore is a small repair shop that makes customized parts for old equipment. All customer orders must be machined first, then polished. Determine a sequence that will minimize the time required to process all six jobs. Chart the schedule on a Gantt chart and indicate the makespan.

17-15. Precision Painters, Inc., has five house painting jobs in one neighborhood. The houses differ in size, state of repair, and painting requirements, but each house must be prepped (cleaned, old paint chipped off, primed) first, then painted. In what sequence should the houses be worked on in order to finish all five houses as soon as possible? Chart out your sequences on Gantt charts and calculate the makespan.

17-16. Sequence the houses in Problem 17-15 by SPT and LPT. How much time is saved with Johnson's rule?

17-17. Tracy has six chapters on her desk that must be typed and proofed as soon as possible. Tracy does the typing; the author does the proofing. Some chapters are easy to type but more difficult to proof. The estimated time (in minutes) for each activity is given here. In what order should Tracy type the chapters so that the entire batch can be finished as soon as possible? When will the chapters be completed?

17-18. Updike Upholstery cuts and sews fabric for custom ordered chairs, ottomans, and sofas. Often, the more complicated patterns are for the smaller pieces, where cutting is more time consuming than sewing. Thus, cutting and sewing times vary. Today's list of jobs, shown below, are for an important customer who needs them shipped out (in one shipment) as soon as possible. Determine the sequence of jobs that will complete the customer's order as quickly as possible, and notify the customer when the order is expected to ship.

17-19. The following data have been compiled for an input/output report at Work Center 7. Complete the report and analyze the results.

17-20. The input /output report for Work Center 6 is as follows. Complete the report and comment on the results.

17-21. Kim Johnson, R.N., the charge nurse of the antepartum ward of City Hospital in Burtonsville, Maryland, needs help in scheduling the nurse workforce for next week.

17-22. Rosemary Hanes needs help in scheduling the volunteers working at the local crisis pregnancy center. Create a work schedule that will meet the demand requirements, given that a volunteer will only work four days per week.

17-23. Schedule the wait staff at Vincent's Restaurant based on the following estimates of demand. Each employee should have two or more days off per week.

17-24. Casey Belzer runs a small machine shop that fabricates parts for sprayers used in foam insulation equipment. With the renewed interest in green building practices and high energy costs, demand for his products have increased dramatically. The shop has three CNC machines that can serve a variety of purposes. As customer orders come in, a routing sheet is developed and the order is diagrammed as shown below. When demand was low, it didn't really matter how the jobs were scheduled. Now, Casey wants to finish each job as quickly as possible so he can move on to the next one. Help Casey develop a schedule that would finish a customer order for 200 units of part A as soon as possible. Assume one B, C, and D are needed for each A.

America Reads, America Counts

America Reads & America Counts (ARC) is a non-profit organization that matches college students with public schools who need support in developing literacy and math skills in the classroom. University students receive federal work study funds for working one-on-one with at-risk children, normally for 10 hours a week in 2-hour increments. The program is most popular in urban areas where there is a large concentration of college and university students. For one university alone in the New York city area, 1000 students are placed each semester in more than 100 elementary and secondary schools.

Placement must consider the academic calendar, changing class schedules, travel distances from schools, student skills (e.g., bilingual) and preferences (grade levels, etc.), and school needs. The process of matching students with schools can take one to two months, by which time applicants may have found other employment and the process must be repeated. Currently the administrator uses an Excel spreadsheet to organize the relevant data, but the assignment process is basically manual. ARC would like you to review their scheduling process and develop a quantitative model that would improve both the speed and quality of assignments. Sample data is given below for your analysis. Assign one student to each school.

From a Different Perspective

"And do you have the answer to Problem 6, Pete?" asked Professor Grasso.

"Yes sir, I have the answer according to the textbook, but I'm not sure I get it," replied Pete.

"You don't understand how to get the solution?"

"Oh, I understand the numbers, but I don't know what they're good for. Where I work, nobody ever 'sequences' anything. You don't have time to calculate things like slack and critical ratio. You do what's next in line or on top of the stack, unless you see a red tag on something that needs to be rushed through. Or maybe you run what's most like what you've just finished working on so the machine doesn't have to be changed. Or you run what can get done the fastest because when you produce more you get paid more."

"Pete, it sounds to me like you are using sequencing rules—FCFS, highest priority, minimum setup, and SPT."

Pete hesitated. "Maybe you're right, but there's still something that bothers me. If you're going to go to all the trouble to rearrange a stack of jobs, you'd want more information than what we're working with."

"What do you mean?"

"I mean, there's no use rushing a job at one station to let it sit and wait at the next. It's like those maniacs who break their neck to pass you on the road, but they never get anywhere. A few minutes later you're right behind them at a stoplight."

"I see."

"You need some way of looking at the entire job, where it's going next, what resources it's going to use, if it has to be assembled with something else, things like that."

"You've got a point, Pete. Why don't you give us a 'real' example we can work with? You talk, I'll write it on the board."

Pete talked for about 10 more minutes, and when he was finished, Professor Grasso had the following diagram on the board.

"Okay, class, let's take this home and work on it. A, B, C, and D are products that comprise a customer's order. They must all be completed before the order can be shipped. The circles represent operations that must be performed to make each product. We've labeled them A1 for the first operation of product A, A2 for the second operation, and so on. The numbers inside the circles are the machines that are used to perform each operation. We have only three machines 1, 2, and 3. Your job is to decide the sequence in which the products should be processed on each machine. There is no setup time between processes, no inventory on hand, and nothing on order. Assume the customer has ordered 50 units of each product. We'll use a process batch of 100 units and a transfer batch of one. Make a Gantt chart for each machine to show us how quickly you can ship the customer's order. Earliest shipment gets 5 extra points on the final exam."

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