Scheduling method and system
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is a continuation of U.S. application Ser. No. 07/077,732 filed Jul. 24, 1987, now abandoned. 
    
    
     TECHNICAL FIELD 
     This invention relates generally to a method and apparatus for sequencing and scheduling fabrication and assembly tasks in a manufactory (e.g., job shop, construction project, or flexible manufacturing facility). 
     BACKGROUND ART 
     A manufactory typically comprises many work centers and workers. Each work center is equipped to perform a certain function. For example, at one work center drilling may be performed, and at another work center cutting may be performed. Furthermore, each worker in the manufactory may be qualified to perform only certain types of functions. For example, one worker may be qualified only to paint, and another worker may be qualified both to paint and to drill. 
     Generally, a manufactory can produce a variety of finished products in response to customer orders. The manufacturing of each finished product typically is divided into many tasks. Tasks, and the order in which they must be performed, may vary from product to product. The tasks of various finished products can be in production simultaneously. This type of manufactory is called a &#34;job shop.&#34; 
     The problems associated with the scheduling of tasks within a job shop have long been recognized in the art. Many managers are unable to develop workable schedules because they have no means of accurately determining the &#34;queue time.&#34; The queue time is the time a task waits because resources (e.g., workers and work centers) are occupied with other tasks. Without accurate schedule information, a manager has no means of assessing and making realistic due date commitments for prospective customer orders. Consequently, workers frequently are reassigned or asked to work overtime to compensate for unanticipated bottlenecks. These disruptions are costly. 
     Existing scheduling techniques attempt to overcome these disruptions by using planned queue times, which are a manager&#39;s estimates of queue times, and time buckets, which are fixed periods of time (typically a week) into which the planning horizon is divided. Based on the planned queue times, each task is assigned to a time bucket. If all tasks assigned to a time bucket cannot be completed during that time bucket, certain systems (called &#34;finite loading&#34;) reassign tasks to other time buckets; other systems (e.g., MRP and MRP II) do not attempt to make these changes. 
     These approaches to scheduling result in infeasible and unrealistic schedules for several reasons. First, the planned queue times are typically very inaccurate. Second, the time buckets must be at least as long as the time to do the longest task and in most cases they are much longer. Third, only one of the two important resources, workers and work centers, is considered in the schedule. Fourth, since either workers or work centers are assigned to tasks before the schedule is developed, unanticipated bottlenecks frequently arise on over-committed resources. Fifth, since revenues and costs are not explicitly considered, the schedule objectives are unrealistic. Finally, since the computer time necessary to develop a schedule is typically long, revisions are infrequent and the schedule does not reflect current production conditions. 
     Accordingly, there is a need for a scheduling system that develops accurate, feasible, minimum cost schedules in a timely manner that allows revisions to reflect current production conditions. 
     DISCLOSURE OF THE INVENTION 
     This invention provides a method and apparatus for cost-based resource scheduling. The object of this invention is to provide a scheduling system that (a) schedules tasks, workers, and work centers and preferably minimizes costs, both inventory carrying and late delivery; and (b) whose efficiencies allow for frequent rescheduling where desired. 
     This, and other objects of the invention, which will become more apparent as the invention is more fully described below, are obtained in preferred embodiments, by a data processing method which first generates an Early Finish Schedule, second generates a Late Finish Schedule from the Early Finish Schedule, and finally generates a Final Schedule by applying a Maximum Flow Procedure (MFP) to a graphic representation of the Late Finish Schedule. 
     In preferred embodiments, the methods and system of the present invention include the following characteristics and advantages: (a) the Early Finish Schedule is a feasible sequence of tasks such that the tasks finish as early as possible; (b) the start time of some tasks in the Early Finish Schedule may be delayed without incurring additional late delivery costs; (c) the inventory carrying costs and the late delivery costs are analogized to the flow of a network in which a Maximum Flow Procedure identifies tasks that can be profitably delayed; (d) the completion times of delayed tasks are adjusted incrementally; and (e) the ordered data structures are represented by heaps. 
     Preferred embodiments rely in part on a network model of the Late Finish Schedule to which is applied the Maximum Flow Procedure. The network model is depicted by an acyclic directed graph, representing initially the Late Finish Schedule. Each node of the graph corresponds to a task in the Late Finish Schedule, and each directed arc corresponds to a precedence requirement. The flow MFP sends through this network corresponds to inventory carrying and late delivery costs. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a flow chart of Phase I (of three phases) of a preferred embodiment of the present invention. 
     FIG. 2 is the first of two flow charts, which comprise Phase II of a preferred embodiment of the present invention. 
     FIG. 3 is the second of two flow charts, which comprise Phase II of a preferred embodiment of the present invention. 
     FIG. 4 is the first of two flow charts, which comprise Phase III of a preferred embodiment of the present invention. 
     FIG. 5 is the second of two flow charts, which comprise Phase III of a preferred embodiment of the present invention. 
     FIG. 6 is the compatibility chart for the illustrative example. 
     FIG. 7 is the Technological Precedence Graph for the illustrative example. 
     FIG. 8 is a table of mnemonics for the illustrative example. 
     FIG. 9 is the Early Finish Schedule for the illustrative example. 
     FIG. 10 is the Schedule Precedence Graph for the illustrative example. 
     FIG. 11 is the Late Finish Schedule for the illustrative example. 
     FIG. 12 is the graph that represents the first Candidate List for the illustrative example. 
     FIG. 13 is the graph that represents the first Candidate List after the MFP was invoked for the illustrative example. 
     FIG. 14 is the Final Schedule for the illustrative example. 
    
    
     BEST MODE FOR CARRYING OUT THE INVENTION 
     This system, the Scheduling System (SS), considers the current state of all the tasks. Each task may be completed, in process, or not started. The SS schedules only those tasks that are not started. All subsequent mention of tasks refer only to those not yet started. 
     The SS represents each queue data structure as a heap. The term &#34;queue&#34; means a sorted list and not a first-in-first-out (FIFO) data structure. A characteristic of a heap is that the first entity in the queue is stored at the top of the heap. The SS generally removes only the first entry in the queue. The heap structure handles revisions (insertions and deletions) efficiently. Each time the SS adds an entry or removes an entry from a queue, the SS adjusts the heap to ensure that the first entry is at the top. Although this embodiment represents the queues as a heap data structure, other data structures may be similarly suitable. These data structures include balanced trees, leftist trees, 2-3 trees, p-trees, and binomial queues. 
     The SS is divided into three phases. Phase takes a set of tasks, workers, and work centers and develops an Early Finish Schedule. Phase II takes the Early Finish Schedule and develops a Late Finish Schedule. Phase III takes the Late Finish Schedule and develops the Final Schedule. 
     This preferred embodiment of the present invention can be best understood with reference to the following terms. 
     DEFINITION SECTION 
     Task Start Time--means the time that a worker is scheduled to start a task at a work center. 
     Task Process Time--means the time that it will take a worker to complete the task. 
     Task Completion Time--means the time when the worker assigned to the task completes the task. The task completion time is equal to the task start time plus the task process time. 
     Prerequisite Tasks--means those tasks that must be completed before the task can start. 
     Task Available Time--means the time when all of the task&#39;s prerequisite tasks are complete. 
     Worker Available Time--means the time when the worker completes his current task. 
     Work Center Available Time--means the time when the worker assigned to the work center completes his current task. 
     Task Available Time is determinable--means that all the task&#39;s prerequisite tasks are either completed or in process. 
     Available Task--means a task that has all of its prerequisite tasks completed. 
     Assigned Worker--means a worker who is currently working on a task. 
     Available Work Center--means a work center that has no worker currently working at it. 
     Worker and Work Center are compatible--means that the worker is qualified to work at the work center. 
     Task and Work Center are compatible--means that the task can be performed at the work center. 
     Task, Worker, and Work Center are compatible--means that the task can be performed by the worker at the work center. 
     Work Center Task Queue--means a queue that holds a list of all available tasks that are compatible with the work center. Each work center has a work center task queue. 
     Schedule Time--means a variable that during scheduling keeps track of the time. 
     End-product Task--means the last task performed in the production of a product. 
     Intermediate Task--means all tasks that are not end-product tasks. 
     Task Priority--means a priority assigned to a task based upon the needs of the manufactory implementing the present invention. A commonly used priority rule is called the minimum slack time rule. It represents the difference between the amount of time until the promised delivery date (Due Date) and the shortest time necessary to complete the task and all tasks on the product which cannot be started until that task is completed. Other priority rules could be used. 
     Phase I 
     The processing of Phase I is represented by the flow chart in FIG. 1. 
     In block 100, the SS initializes the Task Queue. The Task Queue is a list of all not started tasks whose task available time is determinable. The Task Queue is sorted from the earliest to latest task available time. 
     In block 101, the SS initializes the Labor Queue. The Labor Queue is a list of all assigned workers. The Labor Queue is sorted from the earliest to the latest worker available time. 
     In block 102, the SS determines whether both the Task and Labor Queue are empty. If both the queues are empty, Phase I is finished and the SS proceeds to Phase II in FIG. 2. 
     In block 103, the SS compares the available time of the first task in the Task Queue to the available time of the first worker in the Labor Queue. If the task&#39;s available time is less than or equal to the worker&#39;s available time, then the SS removes and selects the first task from the Task Queue, sets the schedule time to the task&#39;s available time, and continues at block 106; otherwise, the SS removes and selects the first worker from the Labor Queue, sets the schedule time to the worker&#39;s available time, and continues at block 104. 
     In block 104, the SS determines whether an available work center exists such that (1) the available work center and selected worker are compatible and (2) the available work center&#39;s Work Center Task Queue contains a not started task. If such a work center does not exist, then SS records the worker as available and continues at block 102; otherwise, the SS selects the work center and continues at block 105. 
     In block 105, the SS selects and removes the highest priority, not started task from the selected work center&#39;s Work Center Task Queue. With this selected task, the selected worker, and the selected work center, the SS continues at block 108. 
     In block 106, the SS determines whether an available work center and an available worker exist such that the selected task, the available worker, and the available work center are compatible. If such a worker or work center does not exist, then the SS continues at block 107; otherwise, the SS selects the worker and the work center, and with the selected task, the selected worker, and selected work center, the SS continues at block 108. 
     In block 107, the SS inserts the selected task into the Work Center Task Queue of all compatible work centers, and the SS continues at block 102. 
     In block 108, the SS schedules the selected worker to start the selected task at the selected work center at the schedule time, as set in block 103. In block 109, the SS records the selected task&#39;s completion time, the selected worker&#39;s available time, and the selected work center&#39;s available time. In block 110, the SS inserts the selected worker into the Labor Queue. In block 111, the SS determines whether the task available time of each immediate successor task of the selected task became determinable when the selected task was scheduled in block 108. For each such immediate successor task, the SS sets the task&#39;s available time to the completion time of the task&#39;s latest completing immediate predecessor task. In block 112, the SS inserts each immediate successor task of the selected task whose available time is determinable into the Task Queue and continues at block 102. 
     Phase II 
     The processing of Phase II is represented by the flow charts in FIGS. 2 and 3. In Phase II, the SS develops the Late Finish Schedule from the Early Finish Schedule. 
     In block 201, the SS develops the Schedule Precedence Graph for the Early Finish Schedule. The Schedule Precedence Graph comprises nodes, which correspond to tasks, and directed arcs, which reflect the order in which tasks must be worked. This ordering of the tasks represents the precedence requirements, which are either technological or assigned. A technological precedence requirement arises from engineering considerations, usually described in a detailed bill of material for the product, whereas an assigned precedence requirement arises from the I sequence of task assignments each worker or work center receives in the Early Finish Schedule. For example, if the Early Finish Schedule assigns a worker to do task A first and task B second, the decision to have the worker do the tasks in that order would appear in the Schedule Precedence Graph as a directed arc from node A to node B. Each node in the Schedule Precedence Graph carries the following information: the worker and work center assigned to the task, the Early Finish Schedule&#39;s task completion time, pointers to immediate predecessor and successor nodes, and the task&#39;s unit delay cost. 
     When an end-product task is completed, the product is finished. A finished product can be delivered on or after its due date. Since delivery prompts payment from a customer (and possibly loss of goodwill if the product is delivered late), end-products have a positive unit delay cost. The positive sign reflects an increase in cost due to revenue from interest on a customer&#39;s payment that is foregone if an end-product task is delayed beyond its due date, less any savings from delaying expenditures for labor and material for that end-product task. 
     Conversely, intermediate tasks do not immediately generate revenue. Since the intermediate tasks require labor and material, but have no offsetting revenues, they have a negative unit delay cost to reflect the savings that their postponement would produce in work-in-process inventory costs. 
     A task&#39;s supply is set equal to its unit delay cost. A task with a supply greater than zero is called a facility; all other (intermediate) tasks are activities. 
     In block 202, the SS creates a fictitious node for each end-product task that Phase I scheduled for completion before its due date. Each fictitious node is initialized with a zero process time, a completion time equal to the end-product task&#39;s due date, and a supply equal to the opportunity costs of a customer&#39;s delayed payment. 
     In block 203, the SS adds a directed arc from the end-product node to the fictitious node. The fictitious nodes represent the delivery of those end-product tasks initially scheduled before their due dates. The former end-product tasks are now considered intermediate tasks (activities). In block 204, the supplies of the former end-product tasks are set to reflect any savings from delaying expenditures for labor and material. 
     In FIG. 3, the SS reduces the work-in-process costs by rescheduling intermediate tasks without further delaying any delivery dates. Recall that the task completion times in the Early Finish Schedule are the earliest possible for the selected sequence. These completion times may result in expenses for labor and material that could be deferred to a later date without delaying the delivery of the finished product. The SS reschedules the tasks, whenever possible, to the latest time that still allows the corresponding finished product to be delivered as scheduled in the Early Finish Schedule. Therefore, the SS examines each intermediate task to determine whether it can be rescheduled. 
     In block 301, the SS generates a queue of all activities (intermediate tasks) ordered from their latest to earliest completion times. 
     In block 302, the SS determines whether the queue is empty. If the queue is empty, the SS proceeds to Phase III at FIG. 4; otherwise, the SS continues at block 303. 
     In block 303, the SS reschedules the completion time of the top task in the queue to equal the earliest start time of its immediate successor tasks. 
     In block 304, the SS removes the top task from the queue and continues at block 302. 
     Phase III 
     In Phase III, the SS takes the Late Finish Schedule and develops the Final Schedule. The Final Schedule is a schedule for which no facility tasks can be profitably delayed. A facility task can be profitably delayed when the cost of delaying the task, its supply, can be offset by cost savings by delaying activity tasks. 
     In FIG. 4, the SS starts with the Schedule Precedence Graph and the Late Finish Schedule. Since each task is represented as a node on the Schedule Precedence Graph, the terms &#34;task&#34; and &#34;node&#34; are used interchangeably. 
     In FIG. 4, the SS iteratively generates a Candidate List, which is a list of nodes. The SS then invokes the Maximum Flow Procedure (MFP), which divides the Candidate List into two lists: Stay List and Move List. The Move List contains those nodes that the MFP determines can be profitably postponed. The Stay List contains all the nodes from the Candidate List that are not in the Move List. The SS then delays the start time of the nodes on the Move List through the use of the Move Queue. The Move Queue is a queue of arcs whose tails touch nodes (tail nodes) in the Move List and whose heads touch nodes (head nodes) not in the Move List. The Move Queue is sorted by shortest to longest arc length. The arc length is the time between the completion time of the tail node and the start of the head node. The Move Queue seldom changes substantially from one invocation of the MFP to the next. Therefore, the SS revises the Move Queue rather than generating a new one. 
     Also, the length of an arc on the Move Queue may be shortened with each invocation of the MFP. However, the SS does not update the length of each arc in the Move Queue with each MFP invocation. Nor does it update the start time of each node on the Move List. Rather, the SS only updates the start time of a node when it is returned on the Stay List by the MFP. The SS keeps track of the cumulative delay of the Move List. When a node is removed from the Move List, its start time is delayed by the net change in cumulative delay of the Move List since the node was added to the Move List. When an arc is added to the Move Queue its length is increased by the cumulative delay of the Move List. 
     In block 401, the SS initializes the capacities of each arc in the graph. Each arc has two capacities. The capacity of the arc in the direction of its head is set to zero and of its tail is set to infinity. In the MFP, the SS adjusts the capacities of the arcs to reflect the cost of delaying facility nodes being offset by the savings of delaying activity nodes. The SS initializes the Facility Queue, which is a queue of all facility nodes sorted by earliest to latest start time. 
     In block 402, the SS empties the Move Queue and sets the cumulative delay of the Move List to zero. The SS also empties the Candidate List. 
     In block 403, the SS tests the Facility Queue. If the Facility Queue is empty, then the SS stops and the Final Schedule is complete; otherwise, the SS continues at block 404. 
     In block 404, the SS removes and selects the top node from the Facility Queue. 
     In block 405, the SS tests the selected node. If the selected node has been put on the Candidate List at least once before, then the SS continues at block 403; otherwise it continues at block 406. 
     In block 406, the SS adds nodes to the Candidate List. The SS adds the selected node to the Candidate List. Also, all nodes are added to the Candidate List such that (1) if the start time of the node were delayed it would cause a delay in the start time of the selected node or (2) if the start time of the selected node were delayed the start time of the node would also be delayed. 
     In block 407, the SS invokes the MFP at FIG. 5 and continues when the procedure returns at block 408. 
     In block 408, the SS updates the Move Queue with information returned from the MFP, the Move List and Stay List. The SS adds to the Move Queue those arcs, not already in the Move Queue, whose tail node is in and whose head node is not in the Move List; and the SS increases the length of these arcs by the cumulative delay of the Move List. The SS removes from the Move Queue those arcs whose tail nodes are in the Stay List; and the SS increases the start time of the Stay List nodes by the change in cumulative delay since the node was placed in the Move List. 
     In block 409, the SS tests the Move Queue. If the Move Queue is empty the SS continues at block 402; otherwise the SS continues at block 410. 
     In block 410, the SS sets the Candidate List to contain only those nodes that are in the Move List. The SS removes the top arc, the one with the smallest length, from the Move Queue and selects the head node of the removed arc. Also, the SS resets the cumulative delay of the Move List to equal the length of the removed arc. 
     In block 411, the SS tests the selected node. If the node has not been considered (labeled) by the MFP then the SS continues at block 406; otherwise the SS continues at block 412. The labeling process is more fully described in the explanation of FIG. 5. 
     In block 412, the SS backtracks from the selected node. Since the selected node has been previously labeled by the MFP, the SS can backtrack from the selected node to the starting node for that path. The SS places each of the nodes along the path onto the Candidate List. 
     In block 413, the SS invokes the MFP at FIG. 5 and continues when the procedure returns at block 414. 
     In block 414, the SS tests the Stay List returned by the MFP. If the tail node of the removed arc, the arc removed in block 410, is in the Stay List, then the SS continues at block 408; otherwise it continues at block 406. 
     In FIG. 5, the SS identifies the nodes on the Candidate List whose start times can be profitably delayed. This procedure is called the Maximum Flow Procedure (MFP). The MFP returns the Move List, which is a list of nodes whose completion times can be profitably delayed. 
     In block 501, the SS initializes the Go Queue with all the facility nodes on the Candidate List ordered by latest to earliest completion time. The SS also empties the Stay List and Move List. 
     In block 502, the SS determines whether the Go Queue is empty. If the Go Queue is empty, then the MFP is complete, the SS places all the nodes in the Candidate List but not in the Stay List on the Move List, and the SS returns to FIG. 4; otherwise, the SS continues at block 503. 
     In block 503, the SS removes and selects the top node from the Go Queue. In block 504, the SS determines whether the selected node is in the Stay List or has a supply less than or equal to zero. If either condition exists, then the SS continues at block 502; otherwise, the SS continues at 505. 
     In block 505, the SS starts a new MFP iteration. The SS sets the MFP backtrack label for the selected node to nil. Each node has a MFP backtrack label associated with it. As the SS proceeds through the MFP, the MFP backtrack labels are set to indicate the path in the Schedule Precedence Graph along which the SS travels. These MFP backtrack labels allow the SS to backtrack, when necessary. The nil backtrack label indicates the starting node in the path. 
     In block 506, the SS determines if there is a node adjacent to the selected node that is eligible. A node is eligible if it has not been labeled on this MFP 10 iteration, if the capacity of the arc in the direction of the adjacent node is greater than zero, and if the adjacent node is not on the Stay List. If such an adjacent node exists, then the SS continues processing at block 507; otherwise, the SS continues at 513. 
     In block 507, the SS determines how much supply it can send to the eligible node; this supply is called &#34;flow.&#34; The SS selects the arc between the eligible node and the selected node. The SS can send all of the supply in the selected node up to the capacity of the selected arc in the direction of the eligible node. As stated above, the capacity of an arc in the direction opposite the arrow is always infinite, while initially the capacity in the direction of the arc is zero but can increase. The SS increases the supply of the eligible node, increases the capacity of the selected arc in the direction of the selected node, decreases the capacity of the selected arc in the direction of the eligible node, and decreases the supply of the selected node by the amount of flow. The SS labels the eligible node by setting the MFP backtrack value of the eligible node to point to the selected node. Finally, the SS selects the eligible node and records the current iteration as the iteration number of this selected node. 
     In block 508, the SS tests the supply of the selected node. If the supply is greater than zero then the SS continues at block 506; otherwise, the SS continues at block 509. 
     In block 509, the SS tests the MFP backtrack label of the selected node. If the label is equal to nil, the SS has backtracked to the starting node, so the SS continues at block 502; otherwise, the SS continues at block 510. 
     In block 510, the SS backtracks from the selected node to the node pointed to in the MFP backtrack label. The node pointed to becomes the selected node. In block 511, the SS tests the supply of the selected node. If the supply is greater than zero, then the SS continues at block 512; otherwise it continues at block 509. 
     In block 512, the supply greater than zero in the selected node means that when the SS last encountered, during this MFP iteration, the selected node, all the supply could not be sent to the eligible node because the capacity of the arc was smaller than the supply of this node. Since the SS could not send all the supply, a residual amount remained. In block 512, the SS removes the label of the selected node and the label of all nodes that were labeled after the selected node. Therefore, these nodes are eligible to receive more supply during this MFP iteration. The SS continues at block 506. 
     In block 513, the SS has selected a node with a supply greater than zero but there are no eligible adjacent nodes. Consequently, the SS backtracks to return the supply. In block 513, the SS tests the MFP backtrack label of the selected node. If the MFP backtrack label is equal to nil, then the SS continues at block 514; otherwise it continues at block 515. 
     In block 514, the SS has backtracked to the starting node and there is still a positive supply, that is, not all the supply was distributed to the other nodes. The SS adds to the Stay List all currently labeled nodes which were selected during this MFP iteration. Note that a label may have been removed in block 512 and possibly restored to a node during an iteration of MFP. 
     In MFP block 515, the SS backtracks from the selected node to the node pointed to by the MFP backtrack label. The SS selects the arc between the selected node and the pointed to node. The SS determines how much supply can be returned to the pointed to node; this supply is called flow. The supply that can be returned is all the supply in the selected node up to the capacity of the selected arc in the direction of the node pointed to. The SS decreases the supply of the selected node, decreases the capacity of the selected arc in the direction of the pointed to node, increases the capacity of the selected arc in the direction of the selected node, and increases the supply of the pointed to node by the amount of flow. 
     ILLUSTRATIVE EXAMPLE 
     The following example illustrates the manner by which the method develops a schedule. For purposes of this example, the manufactory comprises three workers, designated as W1-W3, and four work centers, designated as WC-WC4. The FIG. 6 is the compatibility chart, which indicates worker and work center compatibility. The marks (X) on the chart indicate which workers can work at each work center (e.g., W1 and W2 can work at WC4). 
     For this example, there are four work orders, designated as WO1-WO4, that are available to be scheduled. In FIG. 7, the technological precedence for these four work orders is shown. Each circle corresponds to a task, designated as T1-T5 within each circle, that must be performed, and each arc corresponds to an ordering of the tasks (e.g., T5 of WO3 cannot be started until both T3 and T4 complete). Within each circle is the work center at which the task is to be performed (e.g., T1 of WO2 can be performed at WC1 or WC3, whereas T2 of WO2 can only be performed at WC2). Also within each circle is the process time for the task (e.g., T1 of WO4 will take three units of time from start to finish). 
     FIG. 8 defines some mnemonics that are used in presenting the schedule. 
     Initially, the tasks are in various stages of completion; i.e., some are complete, some are in process, and some are not started. For this example, the first task of work order one (WO1:T1) is complete; the first task of work order two (WO2:T1) has been in process for one time unit; and all other tasks have not been started. The SS schedules only those tasks that are not started. 
     The preceding has defined the data that is needed to produce the Early Finish Schedule for this example. The method illustrated in FIG. 1 will take this data and will generate the schedule shown in the FIG. 9. Each worker is scheduled to work on the assigned task at the assigned work center starting at the assigned time. For example, worker one (W1) is occupied with the in process task WO2:T1; W1 becomes available for tasks that are not yet started at time two. As another example, worker three (W3) is scheduled to work on the fourth task of work order two (WO2:T4) at work center three (WC3) starting at time five. Prior to starting the task, worker three (W3) is scheduled to be idle from time three to time five, at which time he starts the task. 
     After the Early Finish Schedule has been generated, the method proceeds to FIG. 2, Phase II, which generates the Schedule Precedence Graph. The Schedule Precedence Graph for this example is shown in FIG. 10. This graph is similar to the graph shown in FIG. 7. The differences are that the tasks either in process or already completed prior to the start of scheduling are not shown (i.e., WO1:T1 and WO2:T1), that the fictitious nodes WO1:T3 and WO3:T6 have been added for the two early end-product tasks WO1:T2 and WO3:T5, and that the assigned arcs have been added (e.g., the arc from WO1:T2 to WO2:T2). The assigned arcs illustrate the sequence in which each worker and work center will perform the assigned tasks. For example, the arc from node WO4:T1 to node WO2:T4 reflects that worker three (W3) and work center three (WC3) are scheduled to perform task WO4:T1 and them task WO2:T4 the Early Finish Schedule. The arc from WO1:T2 to WO2:T5 indicates that work center 4 (WC4) is scheduled to perform task WO1:T2 and then task WO2:T5 on the Early Finish Schedule. Additionally, the process time for each task and the due date for each end-product task is shown. 
     The method illustrated in FIG. 3, Phase II, uses this Schedule Precedence Graph to generate the Late Finish Schedule. The task, worker, and work center assignments are not changed from the Early Finish Schedule in FIG. 9. Rather, only the time that a worker is scheduled to start a task is delayed. These delays are such that no end-product task is further delayed beyond its due date. FIG. 11 shows the Late Finish Schedule for this example. 
     The method illustrated in FIG. 4, Phase III, uses the Schedule Precedence Graph and the Late Finish Schedule. As in Phase II, the task, worker, and work center assignments are not changed from the Early Finish Schedule in FIG. 9. Rather, only the time that a worker is scheduled to start a task is delayed. End-product tasks can be delayed past their due date if the cost can be offset by savings in inventory carrying cost. FIG. 12 shows a portion of the Schedule Precedence Graph that represents the first Candidate List. The nodes in the Candidate List are shown along with the arcs such that the start time of the head node equals the completion time of the tail node (e.g., the start time of WO2:T5 is 7; the completion time of WO4:T3 is also 7; since these times are equal, the arc between the nodes is illustrated). In addition, each node in FIG. 12 displays its supply. 
     FIG. 13 illustrates the Candidate List portion of the Schedule Precedence Graph after the MFP was invoked the first time in FIG. 4. The number at the tail of each arc represents the capacity in the direction of the arc. The capacity in the direction opposite the arrow is always infinite. Each node with an (M) below it is in the Move List after the first invocation of the MFP. Since the cost of delaying WO2:T5 can be offset by the savings of delaying WO4:T1, WO4:T2, and WO4:T3, these four nodes are in the Move List. 
     At this stage (block 408 of FIG. 4), the SS identifies those arcs (and their lengths) directed from Move List nodes to nodes not in the Move List. These Move Queue arcs, in order of their lengths, are directed from WO4:T1 to WO2:T4 (with length 1), WO4:T2 to WO3:T2 (with length 1), WO4:T3 to WO4:T4 (with length 2), and WO2:T5 to WO3:T5 (with length 2). 
     The shortest length of these, a length of 1, is the distance the Move List nodes are delayed. The resulting schedule appears in FIG. 14. One more invocation of MFP would confirm that this schedule is the final schedule for the problem. 
     Although the invention has been described herein, primarily with respect to preferred methods and systems, it is not intended that the invention be limited to these particular methods and systems. The invention includes the methods and systems described in the claims which follow, including all legal equivalents. 
     Although the invention is fully described in the &#34;Best Mode for Carrying Out the Invention&#34; section, Tables 1, 2, and 3 contain a hexadecimal listing of an object code embodying the present invention. 
     Table 1 contains an embodiment of Phase I. 
     Table 2 contains an embodiment of Phase II. 
     Table 3 contains an embodiment of Phase III. 
     This object code was generated on a Digital Equipment Corporation VAX 11/780 computer. The VAX computer was operating under VMS Version 4.5. The object code was generated by VAX FORTRAN V2.6-244. 
     The following file assignments were made prior to executing these phases. 
     
         ______________________________________                                    
ASSIGN        INPUT.DAT      FOR001                                       
ASSIGN        PHASE1.DAT     FOR002                                       
ASSIGN        PHASE2.DAT     FOR004                                       
______________________________________                                    
 
    
     The data input format for Phase I is described by the following. The FORTRAN code is the actual statement that reads the data in. The data was stored in a file named INPUT.DAT. 
     
         ______________________________________                                    
INPUT                                                                     
______________________________________                                    
1.  Enter the current time [real variable - floating                      
    format]                                                               
READ (1, *) C                                                             
2.  Enter the number of tasks [integer variable - floating                
    format]                                                               
READ (1, *) N                                                             
3.  Enter the number of work centers [integer variable -                  
    floating format]                                                      
READ (1, *) M                                                             
4.  Enter the number of workers [integer variable -                       
    floating format]                                                      
READ (1, *) Q                                                             
5.  Enter the employee labor rate per unit of time [real                  
    variable - floating format]                                           
READ (1, *) RATE                                                          
6.  Enter the end product goodwill penalty per unit of                    
    time [real variable - floating format]                                
READ (1, *) G                                                             
7.  For each task:                                                        
Enter the work centers that can process the.sub.-- task                   
in order of preference; enter 0 if no work cen-                           
ters remain [integer variable - FORTRAN format                            
(I(J), J = 1, 16)]                                                        
READ (1, 40) (I3(JJ), JJ = 1, 16)                                         
40 FORMAT (16I5)                                                          
Enter a &#34;1&#34;, the task time and raw material value                         
added for the task [real variable - floating                              
format]                                                                   
READ (1, *) VERIFY, DDD, CM                                               
Enter the technological successor for the task;                           
enter 0 if no successors remain [integer vari-                            
ables - FORTRAN format (I(J), = 1, 16)]                                   
READ (1, 40) (I3(I1), I1 = 1, 16)                                         
8.  For each product task:                                                
Enter the due date and revenue for the task [real                         
variables - floating format]                                              
READ (1, *) DDD, PROF                                                     
9.  For each work center:                                                 
Enter the time when the work center will next                             
become available [real variable - floating                                
format]                                                                   
READ (1, *) A                                                             
Enter the last task which was either completed by                         
the work center or is under way at the work                               
center [integer variable - floating format]                               
READ (1, *) LASTWC(j)                                                     
10. For each worker:                                                      
Enter the worker&#39;s current status -- 0 for idle,                          
1 for busy [integer variable - floating format]                           
READ (1, *) STATUSW(k)                                                    
For each busy worker, enter the work center where                         
the worker is occupied [integer variable -                                
floating format]                                                          
READ (1, *) J                                                             
11. For each work center:                                                 
Enter the workers in preferred order who can                              
operate the work center; enter 0 if no workers                            
remain [integer variable - floating format]                               
READ (1, *) K                                                             
______________________________________                                    
 
    
     The Appendix contains object code for an embodiment of Phase I, II, and III. ##SPC1##