Patent Publication Number: US-6904329-B1

Title: Method and apparatus for generating a multi-dimensional cost function

Description:
The United States Government has a paid-up license in this invention and the right in limited circumstances to require the patent owner to license others on reasonable terms as provided for by the terms of Cooperative Agreement No. 70NANB7H3041 awarded by the United States Department of Commerce, National Institute of Standards and Technology (“NIST”), Advanced Technology Program (“ATP”). 

   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   This invention pertains to automated manufacturing environments, such as semiconductor manufacturing, and, more particularly, to a method and apparatus for generating a multi-dimensional cost function. 
   2. Description of the Related Art 
   Growing technological requirements and the worldwide acceptance of sophisticated electronic devices have created an unprecedented demand for large-scale, complex, integrated circuits. Competition in the semiconductor industry requires that products be designed, manufactured, and marketed in the most efficient manner possible. This requires improvements in fabrication technology to keep pace with the rapid improvements in the electronics industry. Meeting these demands spawns many technological advances in materials and processing equipment and significantly increases the number of integrated circuit designs. These improvements also require effective utilization of computing resources and other highly sophisticated equipment to aid, not only design and fabrication, but also the scheduling, control, and automation of the manufacturing process. 
   Turning first to fabrication, integrated circuits, or microchips, are manufactured from modern semiconductor devices containing numerous structures or features, typically the size of a few micrometers. The features are placed in localized areas of a semiconducting substrate, and are either conductive, non-conductive, or semi-conductive (i.e., rendered conductive in defined areas with dopants). The fabrication process generally involves processing a number of wafers through a series of fabrication tools. Each fabrication tool performs one or more of four basic operations discussed more fully below. The four basic operations are performed in accordance with an overall process to finally produce the finished semiconductor devices. 
   Integrated circuits are manufactured from wafers of a semiconducting substrate material. Layers of materials are added, removed, and/or treated during fabrication to create the integrated, electrical circuits that make up the device. The fabrication essentially comprises the following four basic operations:
         layering, or adding thin layers of various materials to a wafer from which a semiconductor is produced;   patterning, or removing selected portions of added layers;   doping, or placing specific amounts of dopants in selected portions of the wafer through openings in the added layers; and   heat treating, or heating and cooling the materials to produce desired effects in the processed water.       

   Although there are only four basic operations, they can be combined in hundreds of different ways, depending upon the particular fabrication process. See, e.g., Peter Van Zant,  Microchip Fabrication A Practical Guide to Semiconductor Processing  (3d Ed. 1997 McGraw-Hill Companies, Inc.) (ISBN 0-07-067250-4). 
   Efficient management of a facility for manufacturing products, such as semiconductor chips, requires monitoring of various aspects of the manufacturing process. For example, it is typically desirable to track the amount of raw materials on hand, the status of work-in-process and the status and availability of machines and tools at every step in the process. One of the most important decisions in controlling the manufacturing process is selecting which lot should run on each process tool at any given lime. Additionally, most machines used in the manufacturing process require scheduling of routine preventative maintenance (PM) procedures and equipment qualification (Qual) procedures, as well as other diagnostic and reconditioning procedures that must be performed on a regular basis, such that the performance of the procedures does not impede the manufacturing process itself. 
   One approach to this issue implements an automated “Manufacturing Execution System” (MES). An automated MES enables a user to view and manipulate, to a limited extent, the status of machines and tools, or “entities,” in a manufacturing environment. In addition, an MES enables the dispatching and tracking of lots or work-in-process through the manufacturing process to enable resources to be managed in the most efficient manner. Specifically, in response to MES prompts, a user inputs requested information regarding work-in-process and entity status. For example, when a user performs a PM on a particular entity, the operator logs the performance of the PM (an “event”) into an MES screen to update the information stored in the database with respect to the status of that entity. Alternatively, if an entity is to be taken down for repair or maintenance, the operator logs this information into the M ES database, which then prevents use of the entity until it is subsequently logged back up to a production ready state. 
   Although MES systems are sufficient for tracking lots and machines, such systems suffer several deficiencies, the most obvious of which are their passive nature, lack of advance scheduling, and inability to support highly automated factory operations. Current MES systems largely depend on manufacturing personnel for monitoring factory state and initiating activities at the correct time. For example, a lot does not begin processing until a wafer fab technician (WFT) issues the appropriate MES command. And, prior to processing, a WFT must issue an MES command to retrieve the lot from the automated material handling system (AMHS) with sufficient advance planning that the lot is available at the process tool when the process tool becomes available. If the WFT does not retrieve the lot soon enough, or neglects to initiate processing at the earliest available time, the process tool becomes idle and production is adversely impacted. 
   These types of deficiencies in the typical automated MES emphasize the importance of the wafer fabrication technician (WFT) in the efficient operation of the manufacturing process. WFTs perform many vital functions. For instance, WFTs initiate dispatching, transport, and processing as their attention and time permits. The make scheduling decisions such as whether to run an incomplete lot, as opposed to waiting for an approaching lots, or performing PM or qualification instead of processing lots. However, the presence of WFTs also inevitably introduces some inefficiencies. Typically, there may be a significant difference between the performance of the best WFT and the performance of the worst WFT. A WFT typically simultaneously monitors the processing of many tools, making it difficult to focus on an individual lot or tool. Furthermore, the size and complexity of the modern fabrication process flows makes it exceedingly difficult for a WFT to foresee and prevent downstream bottlenecks or shortages arising from upstream bottlenecks. Shift changes, rest breaks, and days off for the WFT also create inefficiencies or downtime that adversely impact the manufacturing process flow. Just as the importance of the WFT is magnified by the deficiencies of the automated MES, so are the inefficiencies of the WFT magnified by his importance. 
   Thus, factory control systems utilized in today&#39;s wafer fabs are passive and do not enable a high degree of automation. These systems are very dependent on wafer fab technicians and other factory staff to monitor the state of the factory, to instantaneously react to constant change, to make rapid logistical decisions and to initiate and coordinate factory control activity in a timely manner. These wafer fab technicians are agents, providing the active “glue” that is lacking in factory control systems. As a result, factory effectiveness in the highly competitive semiconductor industry is quite dependent on the availability, productivity, skill level and consistency of these human agents. Wafer fab technicians must monitor and operate a number of tools located in various bays in a fab. They are forced to multiplex across tools, bays, material handling systems and a variety of factory control systems. As a fab&#39;s production ramps and more complex processes are introduced, it is difficult to achieve the scalability required to meet the increased complexity. Wafer fab tech visibility of upstream and downstream operations, tool state, work-in-process and resource availability is limited. 
   However, key logistical decisions are frequently based on this limited and dated information, which is only partially provided by factory control systems. Wafer fab techs spend an inordinate amount of time interacting with systems, performing non-value added functions. Shift changes disrupt the operation of the fab as the technicians are temporarily unable to provide required monitoring and coordination. Despite the best efforts of the technicians, utilization of tools suffers, adversely impacting other key factory metrics including cycle time, inventory levels, factory output and mix. With the need for intrabay material handling to transport 12-inch wafers in new 300 mm wafer fabs, significant additional complexity is introduced. Factory control systems are not capable of providing this level of detailed scheduling and execution control. 
   The present invention is directed to overcoming, or at least reducing the effects of, one or more of the problems set forth above. 
   SUMMARY OF THE INVENTION 
   One aspect of the present invention is seen in a method for generating a cost function for processing a candidate workpiece using a resource. The method includes identifying processing requirements for the candidate workpiece. A first committed capacity of the resource is determined based on a schedule of engagements associated with other workpieces having processing requirements compatible with the processing requirements of the candidate workpiece. A second committed capacity of the resource is determined based on a schedule of engagements associated with other workpieces having processing requirements not compatible with the processing requirements of the candidate workpiece. The cost function is generated based on the first and second committed capacities. 
   Another aspect of the present invention is seen in a system including a resource for processing a candidate workpiece and at least one scheduling agent. The scheduling agent is configured to identify processing requirements for the candidate workpiece, determine a first committed capacity of the resource based on a schedule of engagements associated with other workpieces having processing requirements compatible with the processing requirements of the candidate workpiece, determine a second committed capacity of the resource based on a schedule of engagements associated with other workpieces having processing requirements not compatible with the processing requirements of the candidate workpiece, and generate the cost function based on the first and second committed capacities. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The invention may be understood by reference to the following description taken in conjunction with the accompanying drawings, in which like reference numerals identify like elements, and in which: 
       FIG. 1  is a diagram of a portion of one particular embodiment of a process flow constructed and operated in accordance with the present invention; 
       FIG. 2  is a partial block diagram illustrating selected portions of the hardware and software architectures of the computing devices in  FIG. 1 ; 
       FIG. 3A  is a simplified block diagram of a system including provider and consumer software agents in accordance with the present invention; 
       FIG. 3B  is a simplified flow diagram of a method for scheduling resources that may be employed by the software agents of  FIG. 3A ; 
       FIG. 4  is a simplified block diagram illustrating interactions between specialized scheduling and processing agents adapted to schedule and control processing of workpieces, such as lots of semiconductors, through the process flow of  FIG. 1 ; 
       FIGS. 5A through 5D  illustrate parameterized exponential functions; 
       FIGS. 6A and 6B  are a simplified flow diagram illustrating the overall negotiation strategy used by the lot scheduling agent of  FIG. 4 ; 
       FIG. 7  is a graph of a penalty function; 
       FIGS. 8A-8C  are graphs of exemplary engagement density curves; 
       FIG. 9  is a graph illustrating a rate per time unit function; 
       FIG. 10  is a graph illustrating a flexibility discount function; 
       FIG. 11  is a simplified flow diagram illustrating the computing of a basic cost function; 
       FIG. 12  is a diagram illustrating parameters for defining an engagement density curve; 
       FIG. 13  is a diagram of an engagement density curve for a process tool with multiple scheduled engagements that exceed the capacity of the process tool; 
       FIG. 14  is a diagram of a portion of an engagement density curve used to calculate an area under the curve; 
       FIGS. 15-18  illustrate the effects of shifting or changing the widths of engagements on representative engagement density curves; 
       FIGS. 19-21  illustrate the engagement density area that may be recovered by shrinking engagement windows; 
       FIG. 22  is a diagram illustrating the combining of engagements in a batching optimization; and 
       FIG. 23  is a diagram illustrating the combining of engagements in a setup chain optimization. 
   

   While the invention is susceptible to various modifications and alternative forms, specific embodiments thereof have been shown by way of example in the drawings and are herein described in detail. It should be understood, however, that the description herein of specific embodiments is not intended to limit the invention to the particular forms disclosed, hut on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims. 
   DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS 
   Illustrative embodiments of the invention are described below. In the interest of clarity, not all features of an actual implementation are described in this specification. It will of course be appreciated that in the development of any such actual embodiment, numerous implementation-specific decisions must be made to achieve the developers&#39; specific goals, such as compliance with system-related and business-related constraints, which may vary from one implementation to another. Moreover, it will be appreciated that such a development effort might be complex and time-consuming, but would nevertheless be a routine undertaking for those of ordinary skill in the art having the benefit of this disclosure. 
     FIG. 1  conceptually illustrates a portion of one particular embodiment of a process flow  100  constructed and operated in accordance with the present invention. The process flow  100  fabricates semiconductor devices. However, the invention may be applied to other types of manufacturing processes. Thus, in the process flow  100  discussed above, the lots  130  of wafers  135  may more generically referred to as “workpieces.” The term workpiece may refer to one wafer or one lot of wafers. The process tools  115  and any process-operation performed thereon need not necessarily be related to the manufacture of semiconductor devices in all embodiments. However, for the sake of clarity and to further an understanding of the invention, the terminology pertaining to semiconductor fabrication shall be retained in disclosing the invention in the context of the illustrated embodiments. 
   The illustrated portion of the process flow  100  includes two stations  105 , each station  105  including a computing device  1110  communicating with a process tool  115 . The stations  105  communicate with one another over communications links  120 . In the illustrated embodiment, the computing devices  110  and the communications links  120  comprise a portion of a larger computing system, e.g., a network  125 . The process tools  115  are shown in  FIG. 1  processing lots  130  of wafers  135  that will eventually become integrated circuit devices. The process tool  115  may be adapted to process more than one of lots  130  simultaneously. Such a process tool  115  is referred to as a batching tool. 
     FIG. 2  depicts selected portions of the hardware and software architectures, respectively, of the computing devices  110  programmed and operated in accordance with the present invention. Some aspects of the hardware and software architecture (e.g., the individual cards, the basic input/output system (BIOS), input/output drivers, etc.) are not shown. These aspects are omitted for the sake of clarity, and so as not to obscure the present invention. As will be appreciated by those of ordinary skill in the art having the benefit of this disclosure, however, the software and hardware architectures of the computing devices  110  may include many such routine features. 
   In the illustrated embodiment, the computing device  110  is a workstation, employing a UNIX-based operating system  200 , but the invention is not so limited. The computing device  110  may be implemented in virtually any type of electronic computing device such as a notebook computer, a desktop computer, a mini-computer, a mainframe computer, or a supercomputer. The computing device  110  may even be, in some alternative embodiments, a processor or controller embedded in the process tool  1115 . The invention also is not limited to UNIX-based operating systems. Alternative operating systems (e.g., Windows™-based or disk operating system (DOS)-based) may also be employed. The invention is not limited by the particular implementation of the computing device  110 . 
   The computing device  110  also includes a processor  205  communicating with storage  210  over a bus system  215 . The storage  210  typically includes at least a hard disk (not shown) and random access memory (RAM) (not shown). The computing device  110  may also, in some embodiments, include removable storage such as an optical disk  230 , a floppy electromagnetic disk  235 , or some other form, such as a magnetic tape (not shown) or a zip disk (not shown). The computing device  110  includes a monitor  240 , keyboard  245 , and a mouse  250 , which together, along with their associated user interface software  255  comprise a user interface  260 . The user interface  260  in the illustrated embodiment is a graphical user interface (GUI), although this is not necessary to the practice of the invention. 
   The processor  205  may be any suitable processor known to the art. For instance, the processor may be a general purpose microprocessor or a digital signal processor (DSP). In the illustrated embodiment, the processor  205  is an Athlon™ 32-bit processor commercially available from Advanced Micro Devices, Inc. (AMD), but the invention is not so limited. The 64-bit UltraSPARC™ or the 32-bit microSPARC™ from Sun Microsystems, or any of the Itanium™, Pentium™, or Alpha™-class processors from Intel Corporation might alternatively be employed. 
   Each computing device  110  includes, in the illustrated embodiment, a software agent  265  residing in the storage  210 . Note that the software agents  265  may reside in the process flow  100  in places other than the computing devices  110 . The situs of the software agent  265  is not material to the practice of the invention. Note also that, since the situs of the software agents  265  is not material, some computing devices  110  may have multiple software agents  265  residing thereon while other computing devices  110  may not have any. An automated MES  270 , such as WORKSTREAM™, resides on at least one computing device  110 . 
   The computing devices  110  may also be part of a larger network  125  by a connection over the communications links  120 . Exemplary computing systems in such an implementation include local area networks (LANs), wide area networks (WANs), system area networks (SANs), intranets, or even the Internet. The network  125  employs a networked client/server architecture, but alternative embodiments may employ a peer-to-peer architecture. Thus, in some alternative embodiments, the computing devices  110  may communicate directly with one another. The communications links  120  may be wireless, coaxial cable, optical fiber, or twisted wire pair links, for example. The network  125 , in embodiments employing one, and the communications links  120  are implementation specific and may be implemented in any suitable manner known to the art. The network  125  may employ any suitable communications protocol known to the art, e.g., Transmission Control Protocol/Internet Protocol (TCP/IP). 
   Turning now to  FIGS. 1 and 2 , the software agents  265 , collectively, are responsible for efficiently scheduling and controlling the lots  130  of wafers  135  through the fabrication process. Each process tool  115  represents a resource that may be employed for this purpose. For instance, the process tool  115  may be a fabrication tool used to fabricate some portion of the wafers  135 , i.e., layer, pattern, dope, or heat treat the wafers  135 . Or, the process tool  115  may be a metrology tool used to evaluate the performance of various parts of the process flow  100 . Thus, the software agents  265  are capable of assessing a plurality of resources for subsequent processing of the lots  130  of wafers  135 , allocating the resources represented by the process tools  1115 , and negotiating among themselves for the allocation of those resources for subsequent processing of the lot  130  of wafers  135 . 
   In the illustrated embodiment, the software agents  265  are state aware, and are imbued with specific goals that they autonomously initiate behaviors to achieve. The software agents  265  are implemented as objects in an object oriented programming (OOP) environment, but the invention may be implemented using techniques that are not object oriented. Their behavior is relatively simple and is script or rules-based. The behavior is designed to achieve selected goals such as achieving an assigned lot due dale, achieving a predefined level of quality, maximizing process tool utilization, and scheduling opportunistic preventive maintenance. In furtherance of these objectives, the software agents  265  interface with the MES  270  and are integrated with other existing factory control systems (not shown). As will be apparent to those skilled in the art having the benefit of this disclosure, the manner in which this interface and integration occurs is implementation specific, depending upon the particular makeup and configuration of the MES  270  and the factory control systems. 
   Collectively, the software agents  265  schedule ahead for each lot  130  one or more operations on a specific qualified process tool  115 , including transports and required resources, as discussed further below. This includes making optimizing decisions such as running an incomplete batch, as opposed to waiting for an approaching lot  130 , and scheduling opportunistic preventive maintenance or qualifications tests to meet specifications. The software agents  265  schedule activities such as initiating lot transport and processing, performing MES transactions, monitoring processing and transport, and reacting to deviations from scheduled activities or unscheduled activities. More particularly, the software agents  265  may, for instance:
         schedule and initiate execution of interbay material transport required for a next lot processing engagement at a specified process tool  115 ;   monitor transport activity and react to deviations;   schedule and initiate automated material handling system (AMHS) intrabay transport to a reserved process tool port by a specified time;   detect process tool port carrier arrival via auto-identification or equipment event;   initiate loading, recipe download, processing, and unloading on a process tool  115  via an equipment interface;   perform MES transactions;   monitor processing activity and notify WFTs of abnormalities;   detect near completion of processing via an equipment event;   initiate AMHS intrabay transport to the nearest stocker or a nearby process tool  115 ;   detect carrier departure and release the port;   schedule preventive maintenance procedures and notify maintenance technicians (MTs) at the appropriate time; and   schedule qualification procedures and notify WFTs at the appropriate time.       

   Note that, depending on the level of implementation, a given embodiment may implement any or all of these functions, or even functions riot listed above. 
   Referring now to  FIG. 3A , in a general sense, the software agents  265  can typically be classed as “consumer agents”  305  and “provider agents”  310 . Consumer agents  305  represent the interests of consumers  315 , e.g., the PM procedures in performing preventative maintenance within the allowable windows or the lots  130  in advancing through the process flow  100  in a timely and efficient manner. Provider agents  310  represent the interests of providers  325 , e.g., machines such as the process tool  115 , in meeting the demands of consumers for processing resources in advancing the lots  130  through the process flow  100  in a timely and efficient manner. For instance, a software agent  265  representing a lot  130  of wafers  135  would be considered a “consumer” agent  305  and a software agent  265  representing a process tool  115  would be considered a “provider” agent because the process tool  115  is “providing” processing resources “consumed” by the lot  130 . A software agent  265  may be classed as a provider agent  310  in one context and a consumer agent  305  in another context. 
   As noted above, the distinction between consumer agents  305  and provider agents  310  is particularly apt in the context of scheduling. The scheduling of actions initiated by the software agents  265  revolve around budgets, costs, and ratios associated with the processing. More particularly, to further the implementation of a negotiation protocol for allocating resources, a combination of budgets, costs, and ratios are used to implement a scheduling system. The combination is structured to encourage “desirable” behavior, e.g., meeting due dates, effective utilization of machines, etc. 
   Turning now to  FIG. 3B , there is illustrated a method  330  practiced in accordance with the present invention. The method  330  may be practiced in a variety of embodiments and implementations, a particular one of which is disclosed below. The consumer software agents  305  and provider software agents  310  use a “contract net negotiation protocol” approach to schedule the consumers  315  for the providers  325 . The consumer agents  305  negotiate with provider agents  310  to reserve access for the consumer  315  to the resources of the provider  325 . This reserved access is referred to as an “engagement.” In this particular embodiment, both the consumer agent  305  and the provider agent  310  schedule the engagement. 
   The method  330  begins by providing a budget for the consumer  315  for a particular process resource, e.g., process time on the process tool  215  it next wants to consume, as set forth in box  335 . The budget can be structured to affect the operation of the process flow  300 . For instance, consumers  315  having a higher priority for completing the process flow  300  may be afforded greater budgets to increase their chances of successfully acquiring the desired process resource. In general, the budget depends on the processing time of the process step, lot priority, lateness with respect to meeting its due date, and other factors. In some embodiments, a consumer  315  may provide a budget calculator to the provider  325 . The budget calculator permits the provider  325  to determine the priority of the consumer  315  relative to other engagements previously booked by the provider  325 . 
   The consumer  315  then issues, through its consumer software agent  305 , a bid request for the consumer  315  to acquire the process resource, as set forth in box  340 . In one implementation, the consumer software agent  305  requests bids from all eligible providers  310  on behalf of a consumer  315 . When a consumer software agent  305  requests a bid, it gives the provider software agent  310  pertinent information such as: the earliest time to begin transport from the last machine; the process-operation to be scheduled; the latest completion time acceptable to the consumer  315 ; and the location from which the consumer  315  will be transported to the provider  310 . 
   The provider  325  then, through its provider soft-ware agent  310 , submits to the consumer  315  at least one bid responsive to the bid request, as set forth in box  345 . The bid includes a cost function from which the consumer  315  may determine a cost to acquire the process resource from the provider  325 . The provider  325  attempts to maximize its “profits” by adjusting the prices it offers in its bids. 
   An engagement includes a “commitment window” (CW), which is a time interval during which the provider  325  commits to meet the engagement. A “kernel” (k) is a time period representing the actual time required by the provider  325  to complete the task. A “working window” (WW) is a subset of the commitment window that the provider software agent  310  may use to constrain the engagement to accommodate other engagements and avoid overcommitting its resources. The provider software agent  310  may shift the edges of the working window such that it is smaller than the commitment window. If no shifting is required to accommodate other engagements, the working window remains the same size as the commitment window. 
   The consumer  315 , through the consumer software agent  305 , then selects a submitted bid in light of the end time of the candidate bid and the cost function of the provider  325 , as set forth in box  350 . The selection process is described in greater detail below with reference to  FIGS. 6A and 6B . A particular consumer  315  might accept a more expensive bid to ensure a more timely delivery of the purchased resource. 
   The consumer  315  awards a contract to the provider  325  corresponding to the selected bid, as set forth in box  355 , through the consumer software agent  305 . However, the provider  325  typically is negotiating with several consumers  315 . It is possible that the provider  325  scheduled another consumer  315  in such a manner that the cost to secure the submitted bid has increased. Thus, the provider  325 , through its provider software agent  310 , determines the actual cost of the bid (i.e., based on the current schedule of engagements) and accepts the contract if the consumer  315  can still afford the updated cost. The provider  325  then confirms the awarded contract, as set forth in box  360 . If not, it notifies the consumer  315  who can then select a second bid or begin the bidding process all over again. The consumer  315  may maintain a schedule or calendar  320  of scheduled engagements and transport activities (i.e., to move it to the next process tool  115 ) to facilitate scheduling of engagements beyond the current process step. 
   Thus, decision-making in the process flow  300  is guided by economic forces of supply and demand. More particularly, consumer software agents  305  are designed to acquire resources more or less aggressively depending on selected factors, such as priority or lateness. Provider software agents  310  are designed to provide such resources more or less aggressively depending on a number of factors, such as the level of utilization (“congestion”). Note that these decisions can be manipulated externally though configurable curves supplied for costs and budgets on which the decisions are made. Working in concert, the consumer and provider software agents  305 ,  310  cooperate to advance the consumers  305  through the process flow  300  in a timely and efficient manner. 
     FIG. 4  depicts a portion of a semiconductor fabrication process flow  400  in which these concepts are further illustrated. The process flow  400  implements the contract net negotiation protocol discussed above relative to FIG.  3 A and  FIG. 3B  in one particular embodiment thereof. More particularly, the process flow  400  includes:
         a lot scheduling agent  405 , which is a consumer software agent representing the lot  130  for scheduling purposes;   a machine scheduling agent  410 , which is both a consumer and a provider software agent, depending on the context in which it is operating, representing the process tool  115  for scheduling purposes;   a machine processing agent  415 , which is a provider software agent that takes actions to initiate activities scheduled by the machine scheduling agent  410 ; and   a resource scheduling agent  417 , which is a provider software agent representing a resource  420  that may be needed by the process tool  115  to perform the scheduled activity.       
   The lot scheduling agent  405  tries to minimize costs while staying on schedule. The machine scheduling agent  410  tries to optimize tool utilization while maximizing profits. 
   The lot  130  and resource  420  may also have corresponding processing agents (not shown) to whom the scheduling agents  405 ,  417  pass control when it is time for a scheduled activity to begin. Note that resource scheduling agents  417  can represent other types of resources, e.g., dummy wafers, empty cassettes, wafer fabrication technicians (WFT), maintenance technicians (MT), etc. 
   The agents  405 ,  410 ,  415 ,  417  employ negotiation techniques that apply market dynamics to scheduling decisions. These decisions are generally influenced by a schedule of prices for a variable that encourages some assignments over others. In the illustrated embodiment, such price schedules are generated from a toolkit of parameterized exponential functions (PEFs). In some cases, the parameters of an individual function can be adjusted to generate the desired schedule, while other applications employ a sum of these functions. The following discussion of the parameterized exponential function toolkit is referenced in the subsequent detailed discussions that follow. 
   In general, exponential functions can be characterized as Increasing vs. Decreasing (i.e., inc or dec), and as Concave vs. Convex (i.e., C or V).  FIGS. 5A through 5D  illustrate the four possible combinations of these characteristics. Each of these functions is a function of one variable, x. These functions may be tuned using four parameters:
         The steepness, S, determines how quickly the function climbs or falls.   The x-reference, x 0 , and y-reference, y 0 , define a point the curve must pass through (x 0 , y 0 ).   The asymptote, a, shifts the line y=a that the curve approaches as it flattens out. For a convex curve, a&lt;y 0 , and for a concave curve, a&gt;y 0 .       

     FIG. 5A  illustrates a decreasing convex function defined by the equation:
   decV ( x,s,x   0   , y   0   ,a )= a +( y   0   −a ) e   s(x     0     −x) .  
     FIG. 5B  illustrates an increasing convex function defined by the equation:
   incV ( x,s,x   0   ,y   0   , a )= a +( y   0   −a ) e   s(x−x     0     ) .  
     FIG. 5C  illustrates a decreasing concave function defined by the equation:
   decC ( x,s,x   0   ,y   0   , a )= a +( a−y   0 ) e   s(x−x     0     ) .  
     FIG. 5D  illustrates an increasing concave function defined by the equation:
   incC ( x,s,x   0   ,y   0   ,a )= a+ ( a−y   0 ) e   s(x     0     −x) .  
   Although the functions are described as having exponential form, it is contemplated that other parameterized function forms may be used to generate cost functions having similar characteristics (i.e., increasing, decreasing, concave up, concave down, etc.). 
   In some applications, several PEF components of the same or different variables may be added together and the resulting sum normalized to meet various performance requirements. Specifying such a normalized sum requires a list of PEF components. For each normalized sum, the following is specified:
         type (dec or inc, V or C) for each PEF;   parameter values (s, x 0 , y 0 , a) for each PEF;   the variable over which each PEF is computed (x);   the desired maximum value, desMax, of the normalized sum, and, for sums including convex PEFs, the range within which each variable is limited in searching for this maximum; and   the desired minimum value, desMin, of the normalized sum, and, for sums including concave PEFs, the range within which each variable is limited in searching for this minimum.       

   If Max is defined as the maximum of the sum of PEF components within the range of desMax and Min is defined as the minimum of the sum of PEF components within the range of desMin, the formula for the entire function is:
 
desMin+(desMax−desMin)(sum(PEFs)−Min)/(Max−Min) 
 
   Such a composite function may be communicated by sending the type, variable, and four parameters for each PEF and three constants A, B, and C, such that the normalized function is:
 
A+B*(sum(PEFs)−C), where: 
         A=desMin−Min(desMax−desMin)/(Max−Min)   B=(desMax−desMin)/(Max−Min)   C=Min       

   In some cases, a partial function may be passed in a response. In such a partial function the arguments of some of the PEFs are fixed. For example, a bid from a process tool  115  includes a rate per time unit function (RTU) with a fixed ccDiff avg  (described in greater detail below). Such a fixed argument can affect the overall function in two ways. The argument may have its own PEF. In such a case, instead of passing the complete PEF (type, argument, and four parameters), only the value of the PEF at the fixed parameters are passed, and the fixed values are summed together with the other PEFs. This has the same effect as subtracting the value of the PEF for the fixed function from C. In a second case, the fixed argument may need to be summed with a variable argument within the same PEF. In this case, the fixed portion is subtracted from x 0 . Note that whether the argument to the exponential is (x−x 0 ) or (x 0 −x), the effect of replacing x with (x+d) for constant d is the same as replacing x 0  with (x 0 −d). 
   Returning to  FIG. 4 , the negotiation techniques employed by the lot scheduling agent  405  and the machine scheduling agent  410  are described in greater detail. A particular lot scheduling agent  405  may negotiate with a number of pieces of equipment, e.g., process tools  115 . The lot scheduling agent  405  tries to find a process tool  115  that will allow the lot  130  to meet its due date. The goals of the lot scheduling agent  405  are to select a process tool  115  that provides the right type of processing and can support its due date requirements. At the same time, the machine scheduling agent  410  tries to acquire lots  130  for processing in a way that optimizes its utilization. Overall, the goals of the machine scheduling agent  410  are to maximize its overall utilization, respect the relative priority of lots  130 , reduce setup or recipe changes, and optimize its batch size. This collaboration of agent interaction results in the scheduling of a lot  130  on a particular process tool  115  within a specified time window. 
   During the negotiation for a particular process step, each lot  130  is assigned a budget. The lot scheduling agent  405  uses the funds in its budget to secure the resources needed to complete the desired process step, also referred to as a process-operation. The budget for the lot  130  may be determined by a budget tool (not shown), or “Budget Calculator” function, that can be called by the scheduling agents  405 ,  410 ,  417 . Each lot  130  of wafers  135  has a budget for each Process Step (Process-Operation). Each lot  130  has an assigned priority. Typically, priorities are assigned manually, although the priority may be set autonomously in accordance with preprogrammed parameters and classifications. The priority assigned to a lot  130  has a significant influence on its budget. 
   The budget tool receives as inputs an identification of the process step and the ending date/time for which it is called. From these inputs, the budget tool determines a budget for the lot  130  to complete the process step at a specific future time. In some embodiments, the budget may also have a component designed to deal with expiring lots  130 , i.e., lots  130  that need to be processed within a certain time to avoid rework. In some processes, the time between steps becomes important to the outcome of the overall process. If the time between process steps is too long, then additional processing is needed or previous process steps are repeated. Process steps where the previous processing creates expiring lots  130  may have a higher budget that reflects the cost of rework. The budget may grow at a rate that allows the lot  130  to purchase processing before the expiration period. Budget components may also be provided to advance lots  130  that are behind schedule or that are necessary for feeding a downstream bottleneck. 
   In one particular implementation, budgets come in two types-lot budgets and PM/Qual budgets. Whereas the lot budgets are dependent on priority, process time, and various ratios, the PM/Qual budgets depend on primarily on duration and position in the calendaring window. The composite PM/Qual budgets are determined similarly to the composite lot budgets, but with emphasis on duration and position in the calendaring window instead of priority, process time, and the various ratios. 
   The lot scheduling agent  405  begins the negotiation by sending a “request bid” message  425  to all machine scheduling agents  410  representing process tools  115  capable of performing a desired manufacturing operation. At this point, the machine scheduling agent  410  is acting as a provider software agent because the process tool  115  is providing process resources, i.e., processing time. The lot scheduling agent  405  requests bids  435  from all eligible process tools  115  on behalf of a lot  130 . As will be appreciated by those in the art having the benefit of this disclosure, eligibility is primarily predicated on whether a process tool  115  possesses or can acquire the resources needed to perform the process step the lot  130  seeks to enter. When a lot scheduling agent  405  requests a bid  435 , it provides the process tools  115  with the:
         transport start time (TST), or earliest time to begin transport from the last location;   process-operation (PO) and process-step (PS) to be scheduled;   consumer&#39;s latest delivery time (LDT C ), or latest completion time acceptable to the lot  130 ;   identity of the last location or “source” location from which the consumer will be transported to the subsequent process tool  115 ; and   identity of the lot  130  requesting the bid  435 .       

   In some embodiments, the lot scheduling agent  405  may provide a budget calculator to the machine scheduling agent  410  agent. The budget calculator permits the machine scheduling agent  410  agent to determine if the lot  130  can still afford the cost of the engagement when the bid  435  is accepted. 
   A critical ratio may be defined for a lot  130  that indicates the degree to which the lot  130  is ahead, behind, or on schedule. The Critical Ratio, CR, for a time slot that ends at time T e  is defined as:
 
 CR =(Time until Due Date− CRAdjust )/(Standard Cycle Time Remaining*β) 
 
or 
 
  CR =(Due Date− T   e   −CRAdjust )/( SCTR *β);
 
where:
         β=a cycle time compression factor based on lot priority   CRAdust=CRA*DFLTLD+CRB;   SCTR=sum (standard process and queue times of all remaining process-operations);   DFLTLD=expected lead-time for the product (i.e., expected total cycle time);   CRA=configurable control A for CR adjustment; and   CRB=configurable control B for CR adjustment.       

   CRAdjust provides a configurable means to encourage lots  130  to finish early to improve on-time delivery. Unexpected events near the end of the processing sequence can suddenly put a lot  130  behind schedule, and there may not be enough time to recover unless the lot  130  is targeted to finish early. DFLTLD is used to set the Due Date for lot starts, i.e., Due Date=Start Date+DFLTLD. The preceding definition of Critical Ratio, CR, leads to the following conditions:
         CR&gt;1.0 the lot  130  is ahead of schedule;   CR=1.0 the lot  130  is on schedule;   0≦CR&lt;1.0 the lot  130  is behind schedule; and   CR&lt;0 the lot  130  has missed the Due Date and is late.       

   In one particular implementation, the lot scheduling agent  405  calculates the initial value of LDT C  as the time at which the lot  130  will fall behind schedule, i.e., where the target critical ratio CR=1.0. The corresponding LDT C  is determined using the following equation:
 
 LDT   C   =DueDate −( CRA*DFLTLD+CRB )− TargetCR *( SCTR*β ) 
         where:   DueDate the time at which the lot  130  is due to complete the process flow  400 ;   SCTR a sum of the standard process and queue times of all the remaining process-operations;   β a cycle time compression factor based on lot priority;   CRA a configurable control A for CR adjustment proportional to expected cycle time;   CRB a configurable control B for CR adjustment to accommodate disruption near the end of the process flow  400 ;   DFLTLD an expected lead-time for the lot  130 , i.e., expected total cycle time; and   TargetCR a target Critical Ratio for LDT C , a configurable variable whose initial to value is set by lot priority and defaults to 1.0.       

   If the lot  130  is behind schedule the initial value calculated for LDT C  may be in the past or may not allow sufficient time for transport, loading, and processing. The kernel of the commitment window, k, represents the actual process duration for the process-operation. The kernel does not include any setup time, since the lot  130  cannot know at the time of bidding whether or not it will require a separate setup. The machine scheduling agent  410  may therefore calculate a revised minimum LDT, LDT min , that considers the estimated transport time (ETT) for moving the lot  130  to the process tool  115  from its last location, the transport start time (TST), the estimated loading time (ELT), and the kernel as follows:
 
 LDT   min   =ETT+TST+ELT+k.  
 
   In some instances it may be desirable to constrain LDT C  so a lot  130  that is ahead of schedule does not immediately consider alternatives that would cause it to give up all of its lead and immediately regress back to “on schedule.” A configurable parameter (i.e., also referred to as a control knob), CRLoss, that specifies a percentage reduction of the Critical Ratio that is acceptable at a single process-operation may be used. Instead of calculating LDT based on CR=1, LDT is calculated based on a TargetCR using the following equation:
 
 TargetCR=CurrentCR *(1 −CRLoss ). 
 
   This alternate formula is generally only used when the resulting TargetCR&gt;1. If TargetCR&lt;1, then the LDT C  is calculated based on CR=1 and increased to LDT min , if necessary. 
   Referring again to  FIG. 4 , the eligible process tools  115  formulate bids  435  and attempt to maximize their “profits” by adjusting the prices they offer in the bids  435 . As mentioned above, the machine scheduling agent  410  maintains a schedule of engagements. 
   Bids  435  from the machine scheduling agents  410  include the following information:
         BCF=Basic Cost Function for the time window [EST, LDT];   EST=Earliest Start Time for processing=Transport Start Time (TST)+Estimated Transport Time (ETT)+Estimated Loading Time (ELT);   ccSameSetup avg =average committed capacity of engagements of the same setup type within the time window [EST LDT];   RTU ccDiff =Rate per unit time function based on committed capacity, with ccDiffSetup (i.e., committed capacity of engagements with a different setup type) fixed at ccDiffSetup avg , the average committed capacity of engagements not of the same setup type within the time window [EST, LDT]; and   FDF=flexibility discount function.       

   If the process tool  115  is a batching machine (i.e., can process more than one lot  130  simultaneously), the bid  435  also includes:
         ccSameBatch avg =the average committed capacity of engagements of the same batch type within the time window [EST, LDT]; and   RTU ccDiff  is computed with ccDiffBatch fixed at ccDiffBatch avg  as well as ccDiffSetup fixed at ccDiffSetup avg .       

   The use of averages in computing RTU ccDiff  is a simplifying assumption. In some embodiments, such averages may not be used. 
   The BCF, as described in greater detail below, defines the cost of processing per unit time (hourly rate) as a function of the date/time when processing occurs. In some instances the BCF may be represented as a table of x-y values at evenly spaced time intervals (x). 
   The RTU function, as described in greater detail below, defines the cost of processing per unit time (hourly rate) as a function of committed capacity. The RTU is represented as a list of PEFs and normalization parameters with ccDiff avg  fixed as described above. When evaluated, summed, and normalized, these yield the rate per unit time at the specified ccSame avg  and the current ccDiff avg  of the process tool  115 . 
   The FDF, as described in greater detail below, is a PEF specifying the penalty imposed by the process tool  115  for overly narrow commitment windows. Typically, the FDF is a single, unnormalized incV function of the ratio between kernel, k, and commitment window widths. 
   After receiving the bids  435  from the machine scheduling agents  410 , the lot scheduling agent  405  generates a collection of candidate bids for each process tool  115  by sampling the BCF for commitment windows with varying sizes, start times and end times in accordance with a BCF search algorithm. The lot scheduling agent  405  first calculates the maximum and minimum size of the commitment windows to be considered. The theoretical minimum size commitment window is k, the kernel, but practically a minimum window size larger than k should be chosen. Assume β is an externally configurable factor applied to the kernel, k, to determine the minimum size of the commitment window. The minimum and maximum size commitment windows are calculated by:
 
 CW   min =(β+1)* k  for 0&lt;β&lt;K p  
 
 CW   max   =LDT−EST,  
 
where 
 
 K   p =[( LDT−EST )/ k ]−1. 
 
   The BCF search algorithm employed by the lot scheduling agent  405  starts with the largest commitment window, CW max , and gradually reduces the commitment window to CW min . The number of window size samples, S, is determined by a configurable parameter CWSamples. In the illustrated embodiment, the initial value is:
         CWSamples=5.       

   The lot scheduling agent  405  calculates a corresponding shrink factor, α, using the equation: 
         α   =       (       CW   min       CW   max       )       1   /     (     S   -   1     )           ,       
 
where S is the desired number of window size samples.
 
   The BCF search algorithm generates a series of commitment window sizes CW i  between CW min  and CW max  as follows:
 
CW i =CW max  
 
 CW   i   =CW   i-1 *α (for i=2, . . . S) 
         or alternatively,
 
 CW   i   =CW   max *α (i−1)  (for i=1, . . . S) 
       

   For each commitment window size CW i , the BCF search algorithm generates a series of J i  window start times TS i ,j and window end times TE i,j . The number of pairs of (TS i,j , TE i,j ) depends on the window size. For the first window of size, CW max  only 1 pair of starting and ending times is possible. For smaller window sizes, more pairs may be possible. The algorithm for generating the number of these pairs for each CW i  and the starting and ending times of these pairs is described below. 
   A reasonable minimum time shift, S min , of the start/end time is k/2, although larger or smaller values may be used. For each commitment window size, CW i , the maximum time shift, S max , for the window CW i  within the larger interval [EST, LDT] is:
 
 S   max   =LDT−EST−CW   i  
 
 S   min   =k/ 2. 
 
   The number of different window positions, N p , for CW i  is determined using:
 
N p =1 +[S   max   /S   min ] (the division is rounded off to an integer value). 
 
   Next, the minimum shift, S min , is adjusted such that S max  is an exact N p  multiple:
 
 S   min   =S   max   /N   p . 
 
   The starting and ending times for each window of size, CW i , are generated using: 
               TS     i   ,   j       =     EST   +       (     j   -   1     )     *     S   min                     for   ⁢           ⁢   j     =   1     ,     …   ⁢           ⁢     N   p                     TE     i   ,   j       =       TS     i   ,   j       +     CW   i               for   ⁢           ⁢   each   ⁢           ⁢     TS     i   ,   j                 
 
   The combination of varying commitment window sizes with unique start times and end times produces a collection of commitment windows described by start time, end time pairs (TS, TE). 
   The lot scheduling agent  405  calculates a cost for each of the candidate bids associated with the commitment windows described by the pairs (TS, TE). To save computation resources, approximations may be used in calculating the cost. Of course, a more exact and resource intensive computational approach may be used. 
   First, the lot scheduling agent  405  calculates the approximate increase, h, in committed capacity that is caused by a candidate bid. This depends only on the kernel, k, and the size of the commitment window, CW i . Note that the factor, h, is the measure of (in)flexibility for which the FDF computes a penalty.
 
 h=k/CW   i  
 
   Next, the change in the rate of the RTU function, R delta , caused by an increase, h, in committed capacity is estimated. This approximation assumes the RTU is a well-behaved function that is easy to compute (i.e., compared to committed capacity or BCF). The average committed capacities for an engagement are estimated (ccSameSetup e , and ccSameBatch e  if batching) using a window size, CW i , anywhere within the larger window [EST, LDT] based on the average committed capacities, ccSameSetup avg  and ccSameBatch avg  if batching, provided in the candidate bid and the increased density, h, caused by the engagement
 
 ccSameSetup   e   =ccSameSetup   avg   +h  
 
 ccSameBatch   e   =ccSameBatch   avg   +h  
 
   The RTU ccDiff  function is used to calculate the rate for committed capacity ccSameSetup avg  and ccSameSetup e , and ccSameBatch avg  and ccSameBatch e , if batching. Note that the RTU ccDiff  already incorporates the ccDiffSetup avg  and ccDiffBatch avg  for the process tool  115 . The difference, R delta , is computed using:
 
 R   delta   =RTU   ccDiff ( ccSameSetup   e   , ccSameBatch   e )− RTU   ccDiff ( ccSameSetup   avg   , ccSameBatch   avg ). 
 
   Note that for RTUs that are positive monotonic in ccSameX, R delta  is positive. R delta  approximates the change in the cost of the engagement due to the change in committed capacity caused by the addition of the engagement to the existing schedule of engagements for the process tool  115 . This cost adjustment, C delta , is simply the product of the rate change and the size of the commitment window:
 
 C   delta   =R   delta   *CW   i  
 
   The total estimated cost, C e , for adding the engagement starting at time TS and ending at time TE is then: 
         C   c     =     FDF   ⁢           ⁢     (   h   )     *       [       C   delta     +     [     h   *       ∫     t   =   TS     TE     ⁢     BCF   ⁢           ⁢     (   t   )     ⁢           ⁢     ⅆ   t           ]       ]     .           
 
   The cost approximation, C e , should be reasonably accurate even when the kernel ratio, h, is high. Note that if the RTU function is the same for all process tools  115  in the same family, C delta  can be calculated once for each commitment window size, CW i , and then used in the cost estimate of every candidate engagement of size CW i  regardless of the process tool  115 . In this case the lot scheduling agent  405  may wish to evaluate all commitment windows of size, CW i , across all process tools  115  before shrinking the commitment window size and evaluating a new set of candidate engagements. 
   The integral ∫BCF(t)dt is calculated as the area under the BCF curve. For each whole interval, Δt, of the BCF table that falls within the range TS≦t&lt;TE, the contribution to the area is Δt*BCF(t). For the partial interval of the BCF table containing the boundary point TS, the contribution to the area is (t t+1 −TS)*BCF(TS), where t i+1  is the end of the interval containing TS (i.e., t i ≦TS&lt;t i+1 ). For the partial interval of the BCF table containing the boundary point TE, the contribution to the area is (TE−t i )*BCF(TE), where t i  is the start of the interval containing TE (i.e., t i ≦TE&lt;t i+1 ). The integral is approximated as the sum of these contributions in the range TS t&lt;TE. As discussed in greater detail below, if Δt is small enough, BCF(t) can be considered constant within the interval [t i , t i+1 ] and the value of BCF(t) can be approximated as the value of BCF(t i ), where t i ≦t&lt;t i+1 . 
   The lot scheduling agent  405  uses an “objective function” to evaluate the bids  435  formulated above. This objective function has the following form:
 
 F=COL ( BidEndTime )* PO   —   Budget*COLF+BidCost,  
 
where the cost of lateness (COL) is a function of the bid end time (TE) and COLF is a configurable weight. The BidCost is the cost of the engagement, C e . In the illustrated embodiment, the cost of lateness is a decV function of the critical ratio of the end time.
 
   The lot scheduling agent  405  minimizes the value of its Objective Function, F. The lot scheduling agent  405  selects bids  435  according to the minimum cost, F, for bids  435  it can afford with its applicable budget (C e ≦total budget). Note that the lot scheduling agent  405  need only have sufficient budget to pay for the bid cost, C e , but the lot scheduling agent  405  selects bids  435  based on the total cost, F. 
   A method  600  illustrating the overall negotiation strategy used by the lot scheduling agent  405  is shown in  FIGS. 6A and 6B . Referring first to FIG.  6 A:
         a) Request bids  435  from all capable process tools  115  for the process-operation. (Box  605 )   b) Generate candidate bids using a BCF search against each bid returned by each process tool  115 . (Box  610 )   c) Evaluate the candidate bids based on the Objective Function, F. (Box  615 ) The lot scheduling agent  405  considers the Bid End Time, TE, and Bid Cost (i.e., C e ) for each bid and selects a bid using the logic outlined in FIG.  6 B. Referring now to FIG.  6 B:
           1) Sort candidate bids by End Time. (Box  620 ).   2) Calculate the Cost of Lateness function, COL(TE), for each candidate bid. (Box  630 ) COL is calculated once for each Bid End Time, TE since the COL is the same.   3) Calculate the Objective Function, F, for each candidate bid. (Box  635 )   4) Resort candidate bids ascending by the value of F and secondarily ascending by Bid End Time. (Box  640 )   5) Select candidate bid with minimum value of F. (Box  645 ) If more than one candidate bid has the same minimum value of F, the candidate bid with the earliest End Time is selected.
               As the estimated committed capacity, CC e , caused by adding the engagement approaches the maximum process tool capacity it presents a greater challenge for machine scheduling. Such engagements are more likely to result in Regions of Violation (ROV) (i.e., where actual committed capacity exceeds the maximum capacity of the process tool  115 ) and ultimately may cause cancellations if the ROV cannot be resolved.   
               
           d) Returning to  FIG. 6A , after the lot scheduling agent  405  selects a bid  435 , it implements the next phase of the Contract Net protocol by sending an award message  440  to the machine scheduling agent  410  for the process tool  115  associated with the selected bid  435 . (Box  650 )   e) If the lot  130  can still afford the cost of the engagement, and the bid cost has not increased by more than a configurable percentage of the original estimated cost, the machine scheduling agent  410  schedules the engagement and sends a confirmation message  455  for the process tool  115  to the lot scheduling agent  405  and the negotiation for this process-operation is completed. (Box  655 ) The lot scheduling agent  405  pays the process tool  115  the actual cost of adding the engagement. The actual cost may be different than the cost the lot scheduling agent  405  estimated. If the actual cost is higher and the lot  130  does not have sufficient budget to afford the time slot, the machine scheduling agent  410  does not confirm the bid  435 .       

   The lot scheduling agent  405  should take initiative to improve its schedule if it has to select a bid  435  that will not keep it on schedule. The machine scheduling agent  410  may also want to initiate negotiation when openings occur. Machine-initiated negotiation provides a stimulus for the lot scheduling agent  405  to improve its schedule. 
   If the machine scheduling agent  410  does not confirm the selected bid  435  in box  655 , then the lot scheduling agent  405  determines whether it needs to start the bidding over (i.e., return to box  605 ) or select the next best bid  435 . If the bid  435  is not confirmed, the lot scheduling agent  405  compares the number of remaining bids  435  to a configurable control, “RebidThreshold.” If the number of remaining bids  435  is greater than the RebidThreshold, the lot scheduling agent  405  returns to the bid selection process described and selects the next best bid  435  (i.e., returns to box  645  in FIG.  6 B). The lot scheduling agent  405  calculates the objective function F (discussed above) for the new bid  435 . If the value of F has not increased by more than a configurable percentage of the objective function F for the best bid  435 , the lot scheduling agent  405  attempts to confirm the next bid  435 . Otherwise, if the remaining bids  435  are less than the RebidThreshold or the objective function f for the next bid  435  has increased too much, the lot scheduling agent  405  begins the entire process over again by requesting bids  435  from all capable process tools  115  with a wider commitment window [TST, LDT] created by increasing the LDT (i.e., returns to box  605 ). In some embodiments, the lot scheduling agent  405  may be configured to rebid after every confirmation denial by simply setting the RebidThreshold to an arbitrarily high value. 
   When rebidding is required, the new value of LDT for the wider commitment window is calculated by decreasing the Critical Ratio associated with LDT by a configurable percentage CR rebid . Assuming 0&lt;CR rebid &lt;1 and CR old  is the Critical Ratio corresponding to the old value of LDT old  (i.e., CR old  is CR calculated with T e =LDT), then the new Critical Ratio, CR new , and LDT new  are:
 
 CR   new   =CR   old   *CR   rebid ; and 
 
 LDT   new   =DueDate−CR   new *( SCTR*β )−( CRA*DFLTLD+CRB ). 
 
   The rebid algorithm described above is a mechanism intended to prevent the selection of sub-optimal bids  435 , while guarding against excessive communications overhead. Several alternative approaches are possible. The more process tools  115  there are, the more expensive it is to rebid, and the less often one wants to do it. The following strategies may be used in determining whether to rebid.
         Rebid when the remaining bids  435  are older than a predetermined time threshold.   Rebid after a certain number of bids  435  have been rejected. Repeated rejections indicate that the bids  435   s  currently in hand are out of synch with the real world.       

   Returning to  FIG. 4 , the lot scheduling agent  405  may schedule process-operations in advance of the current required process step. A configurable “Lookahead” parameter that specifies the number of operations that should be scheduled in advance may be used. “Hot” lots  130  may have a different “Lookahead” value than normal lots  130 . In some implementations, the lot scheduling agent  405  may dynamically increase the “Lookahead” for a lot  130  as it approaches a batching operation to increase effective utilization of the batching tool. 
   Various exception conditions may require adjustments to current or future engagements. The number and identity of such exceptions is implementation specific. In the illustrated embodiment, for instance, exceptions include, but are not limited to, cancellation, finishing early, scrapping, lot priority changes, lot placed on hold, lot reworking, etc. A number of these exceptions are discussed in greater detail below. These exceptions may cause changes in scheduling, both the engagements already booked and those to be booked. One such exception condition occurs when an engagement is cancelled by a process tool  115 . The machine scheduling agent  410  sends a “cancel contract” message  482  indicating that the process tool  115  is canceling an engagement. The lot  130  is refunded the cost it paid for the engagement plus a “penalty refund” that depends on how much time remains before the engagement. The penalty refund is a value that increases as the delta time (time between current time and the start of the engagement) decreases and is also proportional to the priority of the lot  130  according to a “penalty factor.” The formula for the penalty refund, for a Delta Time (in minutes) is:
 
Penalty Refund= P (Delta Time)* PO   —   budget *(Penalty Factor). 
 
   The Penalty Factor is a multiplier that may be configured based on the type of lot  130 . For example, the penalty factor for a normal lot  130  may be 1.0 and the penalty factor for a hot lot  130  may be 1.25.  FIG. 7  illustrates shows a graph of an exemplary Penalty Function, P(Delta Time). In the illustrated embodiment, the penalty function is a decreasing convex (decV) function that may be constructed by specifying PEF parameters, as described above (see FIG.  5 A). 
   The cancellation approach described above assumes that the lot scheduling agent  405  pays for a bid  435  at the machine scheduling agent  410  confirms the bid  435 , thus requiring the machine scheduling agent  410  to refund the original payment. In an alternative embodiment, the actual payment may be deferred until the engagement starts or completes. Accordingly, it would be unnecessary to refund the original price. Also, the actual price may be reduced if subsequent engagements with the same setup or batch requirements may be scheduled proximate the subject engagement. 
   When the lot scheduling agent  405  receives a cancel contract message  482 , it removes the cancelled engagement and begins rescheduling. In one embodiment, a total rescheduling approach may be used. The lot scheduling agent  405  cancels all of its subsequent engagements by sending a “cancel award” message  483  to all of the machine scheduling agents  410  representing the scheduled process tools  115  and begins the entire scheduling process again. The machine scheduling agent  410  does not pay a penalty refund if the engagement is cancelled by the lot scheduling agent  405 . The machine scheduling agent  410  only pays a penalty refund when it initiates the cancellation. In another embodiment, an iterative rescheduling approach may be used. First, bids  435  are solicited to replace the cancelled engagement. If the cancelled engagement can be replaced without overlapping the next engagement scheduled for the lot  130  then rescheduling is completed. If there is an overlap, a cancel award message  483  is sent to the machine scheduling agent  410  representing the process tool  115  for the next, overlapping engagement. The cancelled engagement is in turn replaced, and any overlap with the next scheduled engagement is identified. The rescheduling process continues until there are no overlaps or there are no more engagements scheduled for the lot  130 . 
   Another exception situation occurs when a lot  130  finishes processing earlier than expected (i.e., prior to the end time of the commitment window). When this occurs, the lot scheduling agent  405  notifies the next process tool  115  that it can arrive earlier. The lot scheduling agent  405  sends an “update commitment window” message  484  to the process tool  115  including the following parameters:
         Process-operation;   EST=Earliest Start Time of current commitment window;   Transport Start Time (TST)—updated based on early finish of prior PO; and   Location where transport will begin.       

   The machine scheduling agent  410  for the process tool  115  returns the new EST of the commitment window as:
 
 EST=TST +Transport Time+Loading Time 
 
   The lot scheduling agent  405  may also choose to rebid the next process-operation with a LDT new &lt;LDT old  for the commitment window. Rebidding allows the lot  130  to preserve the time it has gained by finishing early rather than potentially losing this advantage by allowing more time at the next process-operation. 
   Another exception occurs when all the wafers in a lot  130  are scrapped or “sold” to engineering. The lot scheduling agent  405  needs to cancel all future engagements and stop scheduling. In the event of a partial lot scrap or sale, the kernel size, k, (processing time) for future commitments may change. If the kernel size changes, the lot  130  needs to notify all machines involved in its future commitments. The lot scheduling agent  405  sends a “change contract” message  486  for the contract net protocol. The lot scheduling agent  405  sends the change contract  486  message to the process tool  115  to provide the revised kernel size and possibly a new earliest start time (EST). 
   Other exceptions include placing the lot  130  on hold or reworking the lot  130 . If a lot  130  is placed on hold, the lot scheduling agent  405  cancels all of its future engagements by sending cancel award messages  483  to the machine scheduling agents  410  representing the affected process tools  115  and stops scheduling. When the lot  130  is released from hold, the lot scheduling agent  405  begins scheduling the required process-operations. If a lot  130  requires rework, the lot scheduling agent  405  cancels future engagements by sending a cancel award message  483  to the machine scheduling agent  410  representing each of the process tools  115  and schedules new engagements for the rework operations. 
   In some cases, a lot  130  may be split. If the main lot  130  is terminated after the split, its lot scheduling agent  405  cancels all future engagements by sending a cancel award message  483  to the machine scheduling agent  410  representing each of the process tool  115  and stops scheduling. If the main lot  130  still exists with fewer wafers, the situation is treated like a partial lot scrap as described above. A new lot scheduling agent  405  is created for each of the split lots  130  to begin scheduling for them. In some cases, the product for the new split lot  130  may be different than the product associated with the parent lot  130 . 
   Extra wafers may be bonused or bought into an existing lot  130 . If the processing kernel, k, for any future engagement increases, the lot scheduling agent  405  must either pay a fee or reschedule the engagement. The fee to keep the engagement is the cost of an engagement with the same commitment window and a kernel size equal to the change of the kernel size. The lot scheduling agent  405  requests the change by sending the Change Contract message  486  to the machine scheduling agent  410  representing the process tool  115 . If the new kernel does not fit within the commitment window or the working window the machine scheduling agent  410  may deny the change. In a case where multiple lots  130  are merged, the remaining, larger lot  130  is treated like a bonused lot  130 , and the other lots  130  are terminated and treated like a scrapped lot  130 . 
   Exception conditions may also occur if the due date or priority of a lot  130  is changed. The lot scheduling agent  405  may reschedule its engagements if the new due date is earlier or its new priority is higher. 
   Still referring to  FIG. 4 , the operation of the machine scheduling agent  410  is now discussed in greater detail. The machine scheduling agent  410  attempts to maximize its profits by adjusting the prices it offers in bids  435 . The machine scheduling agent  410  maintains several data structures in order to compute prices and keep track of its engagements. Each engagement is tracked with the following information:
         cws=Earliest Start Time of commitment window;   cwe=Latest Delivery Time of commitment window;   k=kernel, i.e., the actual processing time for the engagement;   wws=Earliest Start Time of the working window; and   wwe=Latest Delivery Time of the working window.       

   Initially, wws=cws and wwe=cwe. The machine scheduling agent  410  may increase wws and reduce wwe unilaterally without violating the cws and cwe negotiated with the lot  130  (i.e., cws wws&lt;wwe cwe). 
   The machine scheduling agent  410  maintains an engagement density curve for each individual engagement.  FIG. 8A  illustrates a first case where the size of the working window is large compared to the size of the kernel (i.e., wws+k&lt;wwe−k). In the density curve of  FIG. 8A , the ratio between the kernel width and the working window width is about 25%. The height of the density curve represents the likelihood that the kernel will be processed by the process tool  115  at the given time. As the width of the working window shrinks compared to the size of the kernel, the height of the trapezoid increases until a boundary condition is reached (i.e., wwe−k=wws+k:ratio=50%), and the density curve becomes triangular, as shown in FIG.  9 B. As the width of the working window continues the shrink (i.e., t e −k&lt;wws+k), the density curve resumes a trapezoidal form, as shown in FIG.  8 C. As seen in  FIG. 8C , the kernel is sure to be executed by the process tool process tool  115  during the plateau region (i.e., probability=1). In the density curve of  FIG. 8C , the ratio between the kernel width and the working window width is about 75%. 
   In the illustrated embodiment, the density curve for each engagement is represented as a piecewise linear function comprised of three segments of the form:
 
 E ( t )= mx+b.  
 
   Each segment has a segment start time, s s , and a segment end time, s e , and each segment is one of an Initial, Final, or Medial segment of the engagement. Thus, a segment of an engagement is stored as an ordered list with an engagement id, eid, and a segment id, seg (I, F, M):
 
Engagement Segment=&lt;eid, seg, s s , s e , m, b&gt;. 
 
   The parameters for the engagement curves for the first case (i.e., wws+k&lt;wwe−k) are summarized below in Table 1. 
   
     
       
         
             
           
             
               TABLE 1 
             
           
          
             
                 
             
             
               Engagement Density Curve Parameters (Case 1) 
             
          
         
         
             
             
             
             
          
             
               Segment 
               Density 
                 
                 
             
             
               ID 
               (t,k,wws, wwe) = 
               Condition 
               Parameters for E(t) = mx + b 
             
             
                 
             
             
               Initial 
               (t − wws)/ 
               wws ≦ t &lt; 
               s s  = wws, 
             
             
                 
               (wwe − wws − k) 
               wws + k 
               s e  = wws + k, 
             
             
                 
                 
                 
               m = 1/(wwe − wws − k), 
             
             
                 
                 
                 
               b = −wws/(wwe − wws − k) 
             
             
               Medial 
               k/ 
               wws + 
               s s  = wws + k, 
             
             
                 
               (wwe − wws − k) 
               k ≦ t &lt; 
               s e  = wwe − k, 
             
             
                 
                 
               wwe − k 
               m = 0, 
             
             
                 
                 
                 
               b = k/(wwe − wws − k) 
             
             
               Final 
               (wwe − t)/ 
               wwe − 
               s s  = wwe − k, 
             
             
                 
               (wwe − wws − k) 
               k ≦ t &lt; 
               s e  = wwe, 
             
             
                 
                 
               wwe 
               m = −1/(wwe − wws − k), 
             
             
                 
                 
                 
               b = wwe/(wwe − wws − k) 
             
             
                 
             
          
         
       
     
   
   The parameters for the engagement curves for the second case (i.e., t e −k&lt;wws+k), are summarized below in Table 2. 
   
     
       
         
             
           
             
               TABLE 2 
             
           
          
             
                 
             
             
               Engagement Density Curve Parameters (Case 2) 
             
          
         
         
             
             
             
             
          
             
                 
               Density 
                 
                 
             
             
               Segment 
               (t,k,wws, 
                 
                 
             
             
               ID 
               wwe) = 
               Condition 
               Parameters for E(t) = mx + b 
             
             
                 
             
             
               Initial 
               (t − wws)/ 
               wws ≦ t &lt; 
               s s  = wws, 
             
             
                 
               (wwe − 
               wwe − k 
               s e  = wwe − k, 
             
             
                 
               wws − k) 
                 
               m = 1/(wwe − wws − k), 
             
             
                 
                 
                 
               b = −wws/(wwe − wws − k) 
             
             
               Medial 
               1 
               wwe − k ≦ t &lt; 
               s s  = wwe − k, 
             
             
                 
                 
               wws + k 
               s e  = wws + k, 
             
             
                 
                 
                 
               m = 0, 
             
             
                 
                 
                 
               b = 1 
             
             
               Final 
               (wwe − t)/ 
               wws + k ≦ t &lt; 
               s s  = wws + k, 
             
             
                 
               (wwe − 
               wwe 
               s e  = wwe, 
             
             
                 
               wws − k) 
                 
               m = −1/(wwe − wws − k), 
             
             
                 
                 
                 
               b = wwe/(wwe − wws − k) 
             
             
                 
             
          
         
       
     
   
   In another boundary case, where the kernel width is the same as the working window width (i.e., wwe−wws=k), the engagement density curve has initial and final segments with zero length and infinite slope. The curve is rectangular with a height of 1 and a width equal to the width of the kernel. 
   The sum of the engagement density curves for the scheduled engagements of the process tool  115  is referred to as the committed capacity of the process tool  115 . A committed capacity curve (CCC) represents the committed capacity as a function of date/time, cc(t). The individual engagements do not include setup time, so the CCC does not reflect process tool capacity that is consumed by setups. This discrepancy may be handled by adjusting the threshold used to determine when the committed capacity is too high, as discussed in greater detail below. 
   The CCC may be represented as a piecewise linear function computed as the sum of the Engagement Density Curves. To compute this curve, observe first that the sum of two line segments m 1 t+b 1  and m 2 t+b 2  is (m 1 +m 2 )t+(b 1 +b 2 ), so that the sum of a set of linear segments has a slope that is the sum of their slopes and an intercept that is the sum of their intercepts. Thus, an ordered list of all the points in time at which a segment either begins or ends is generated. Each successive pair of such points defines a line segment of cc(t) whose slope is the sum of the slopes of all segments that begin at or before the first point and end at or after the second (i.e., with some minor adjustments to be outlined later). 
   One technique for managing the CCC process is to form a table with a separate row for each segment start, s s , and each segment end, s e . Thus, each segment generates two rows in the table. The table has one column for each segment currently scheduled on the process tool  115 . In addition, the initial column time point, tp, is the s s  or s e  that generated the row. The machine scheduling agent  410  processes the table in the following manner:
         1. Sort the table by tp.   2. Delete all but one of any rows with the same tp. Thus, each successive pair of rows corresponds to one linear component in the CCC.   3. For each row, mark each segment&#39;s column with ‘1’ if s s ≦tp&lt;s e , and ‘0’ otherwise. Note that segments with s s =s e  will all be marked ‘0’ by this rule.   4. The slope of the component in the CCC beginning with a given row is the sum of the slopes of the segments marked ‘1’ in that row, and its intercept is the sum of the intercepts of those segments. Since no segment with s s =s e  will ever be marked ‘1’, Initial or Final segments with infinite slopes are excluded from this summation and do not cause any problems. The CCC may have vertical steps, but its value at a time, t, that falls at the junction of two segments is the value of the segment that starts at t, so these vertical steps do not cause problems.       

   Mathematically, this computation can be expressed as a matrix multiplication. Let T be an {r×s} matrix defined using the table described above, without its first column (i.e., containing only the 1&#39;s and 0&#39;s, with the r rows indexed by time and the s columns by segment). Let S be a {s×2} matrix with one row for each segment, containing the segment&#39;s slope in the first column and the segment&#39;s intercept in the second. Then, T×S is a {r×2} matrix whose first column describes the slope of the CCC component beginning at the time represented by that row, and whose second column is the intercept of that component. 
   Each segment is of the form:
 
Committed Capacity= m *Time+ b  
 
   Each segment is represented as:
         T s =starting date/time of line segment;   T e =ending date/time of line segment;   m=slope; and   b=intercept,   where   T s &lt;T e .       

   For batching or setup optimization, a separate CCC is maintained for each type of lot  130 .
         Two lots  130  are of the same type with respect to batching if they could be in the same batch.   Two lots  130  are of the same type with respect to setup optimization if they could share a setup.       

   When constructing a bid  435  for a lot  130  of a given type for a process tool  115  that must support hatching or setup optimization, the machine scheduling agent  410  adds all the CCCs that are not of the type under consideration into a CCC that defines ccDiff (i.e., the percentage of the resource&#39;s capacity over time that is committed to engagements of a different type). The CCC for the lots  130  of the same type is then defined as ccSame. The total utilization is the sum of the same and different components:
 
 cc=ccDiff+ccSame.  
 
   Each ccDiff or ccSame is made up of contiguous segments ccDiff i  or ccSame i , and each segment has its own slope mDiff i  or mSame i  and intercept bDiff i  or bSame i . Segment i applies to all times t i  such that:
 
(start of segment  i )≦ t   i &lt;(end of segment  i ). 
 
   The equations for segments of ccDiff and ccSame are:
 
 CcDiff   i ( t   i )= mDiff   i   *t   i   +bDiff   i ; and 
 
 CcSame   i ( t   i )= mSame   i   *t   i   +bSame   i , 
         where (mdiff i , bDiff i ) are computed by considering all segments that differ in kind from the lot  130  and (mSame i , bSame i ) are computed by considering all segments that are of the same kind.       

   The slope, m, and intercept, b, and their related mDiff and bDiff can be viewed as functions of time whose values are constant over each segment. Thus, the CCC can be represented as:
 
 cc ( t )= m ( t )* t+b ( t ), 
         and similarly for ccSame and ccDiff.       

   If a process tool  115  is being optimized for both batching and setup, four cc curves need be represented: ccSameBatch, ccDiffBatch, ccSameSetup, and ccDiffSetup, each with its own set of segments and associated values for (m, b). 
   The discussion now turns to a more detailed description of the RTU function. As introduced above, the machine scheduling agent  410  uses the RTU function to define the dollar rate per unit of processing time based on the committed capacity of the process tool  115 . RTU is represented as a normalized sum of one or more convex exponential functions. The specific characterization of the exponential functions is application dependent, as the desired bidding behavior may vary depending on the particular implementation. In general, the RTU is designed such that the sum of the individual functions reflects the desired bidding behavior. The entire sum is then normalized to the desired rates. Hence, different embodiments may include different numbers of functions with different parameters. 
   In one embodiment, a simple RTU for a process-operation with no batching and no setup optimization may consist of a single increasing convex exponential of total process tool utilization, U. The increasing nature of the RTU raises the price as the process tool  115  becomes more fully committed. This basic RTU is also the foundation of RTUs for batching or setup optimization, since they must also take account of congestion. 
   To encourage lots  130  of the same type to move close to one another (for batching or setup optimization), several component functions are defined and summed to generate the RTU:
         incV(cc)—the base RTU, to avoid congestion;   decV(ccSame)—to encourage lots  130  of the same type to schedule engagements near each other; and   incV(ccDiff)—to discourage lots  130  of different types from scheduling engagements near each other.       

   In some embodiments, it is useful to use two decreasing functions of ccSame with different steepness parameters. A high steepness parameter gives a sharp drop-off so that once one lot  130  of a given type is scheduled on the process tool  115 , other lots  130  will be strongly attracted. The bottom of such a curve is relatively flat. An additional function with a low steepness parameter helps ensure that the bottom of the curve is sloped, so that as the concentration of lots  130  of a given type grows, new lots  130  are even more likely to be attracted. 
   The result of summing these three functions is a function of two variables, ccSame and ccDiff (i.e., cc is just the sum of these two). Such an RTU function may be graphed as a surface over an x-y plot using a mathematical tool such as Microsoft® Excel® or Mathematica®, and the surface may be manipulated in such a tool to select the correct parameters. The objective of such a characterization step is to get an RTU of the correct shape without focusing on the exact values.  FIG. 9  illustrates an exemplary RTU function as a function of ccSame and ccDiff. 
   To optimize both setup and batch optimization, additional parameterized exponential functions are included in the RTU. The fixed parameters provided below are merely illustrative values that may vary depending on the particular implementation. The RTU includes a congestion factor, batch factors and setup factors. The congestion factor is the same for either Same/Different pair, since either pair sums to the same overall utilization. 
   Congestion Factor:
         incV[ccSameSetup+ccDiffSetup, 10, 1, 100,0]       

   Batch factors:
         decV[ccSameBatch, 10, 0, 100,0] (i.e., to favor same type)   decV[ccSameBatch, 1, 0, 50,0] (i.e., to add a decreasing slope)   incV[ccDiffBatch, 1, 1, 200,0] (i.e., to discourage different type)       

   Setup factors:
         decV[ccSameSetup, 10, 0, 100,0] (i.e., to favor same type)   dec V[ccSameSetup, 1, 0, 50,0] (i.e., to add a decreasing slope)   incV[ccDiffSetup, 1, 1, 200,0] (i.e., to discourage different type)       

   The sum of the congestion batch and setup factors is normalized oil the basis of desired minimum and maximum values, desMin and desMax, described above, and the observed Min and Max values of the raw sum of PEFs. Since the RTU is made up of convex PEFs, the maximum of the sum is meaningful only over a given range of arguments, which is also specified. In the illustrated examples the value of the sum when all arguments are 0 is defined as the Max. Then, the normalized RTU is given by:
 
desMin+(desMax−desMin)*(sum(PEFs)−Min)/(Max−Min). 
 
   A process tool  115  that is underutilized in the near future has little time to find engagements that can use that capacity, and may choose to offer an “urgency discount” during that period to attract engagements. The urgency discount is applied as a multiplicative factor to the BCF. The urgency discount is an increasing function of both cc and time, with a maximum value of 1. However, since cc is a function of time, the urgency discount may be expressed as a function of time alone. 
   The machine scheduling agent  410  computes the urgency discount as the minimum of 1 and the sum of two incC PEFs. The argument of one PEF is cc(t), while the argument of the other is simply t. In general, parameter selection for the PEFs is guided by the factors:
         x 0 =0 for both PEFs   y 0  is chosen for each PEF so that the sum of y 0  for both functions is the maximum discount permitted (e.g., for a 50% discount at t=0 and 0 utilization, y 0 =0.25 on both PEFs)   steepness, s, is greater for the cc(t) PEF than for the PEF driven solely by t.       

   The urgency discount function, UDF(t), used in the illustrated embodiment is:
 
 UDF ( t )=Min(1 , incC ( cc ( t ), 3,0, 0.25, 1)+ incC ( t,  0.1, 0, 0.25, 0.75))  (1) 
 
   Since the machine scheduling agent  410  knows cc(t), it can build the urgency discount into the BCF that it reports to the lot  130 . The urgency discount reflects the cc(t) of the process tool  115  without the new lot  130 , since its function is to attract the lot  130  to the process tool  115  to fill otherwise unused capacity. 
   The machine scheduling agent  410  may wish to give a flexibility discount (or inflexibility penalty) to engagements depending on the flexibility. The lower the ratio of kernel width to total window width for an engagement, the more flexibility the process tool  115  has to avoid conflicts by shifting the kernel. The flexibility factor is defined by:
 
 pWind =kernel width/window width. 
 
   A simple flexibility discount may be implemented by multiplying the BCF by an incV function, such as the one shown in FIG.  5 B. The desired level of flexibility is constrained by an x-factor of the process flow  100 . The x-factor of the process flow  100  is the ratio of a planned cycle time to a theoretical cycle time. The planned cycle time is the expected elapsed time between starting and finishing a lot  130  of the given product and lot priority in the process flow  100  for the product. The theoretical process time is the sum of the processing times, or kernels, for all the process steps in the process flow  100 . In the illustrated embodiment, the planned cycle time is distributed across process-operations, and the average commitment window is the kernel width times the x-factor. Thus, the appropriate pWind value for a given x-factor is just 1/x-factor. For example, if the target x-factor of the process flow  100  is 4, an exemplary FDF may be defined as:
 
 FDF ( pWind )= incV ( pWind,  1.5, 0.25, 1, 0).  (2) 
 
   In an alternative embodiment, each process-operation may have a queue time, which is the expected time a lot waits before processing starts. In such an embodiment, pWind may be calculated by:
 
 pWind =kernel time/(queue time+kernel time). 
 
     FIG. 10  illustrates the shape of this exemplary FDF. In this example, the FDF returns a multiplier of 1 for pWind=0.25 (i.e., where the dashed lines intersect, representing an x-factor of 4). Lots  130  that offer more flexibility than this level (i.e., pWind&lt;0.25) have their costs multiplied by a factor less than 1. Thus, they receive a discount. Conversely, lots  130  that offer less flexibility than this level (pWind&gt;0.25) see their costs multiplied by a factor greater than 1, and thus experience a penalty. 
   The relationship, pWind=1/x-factor, suggests that these penalties and discounts are backwards, since we are giving a discount for pWind less than the target, which corresponds to a higher x-factor. The expected x-factor helps predict what level of flexibility is expected, and enables allocation of flexibility fairly across lots  130 . But the more flexibility a process tool  115  is given, the better throughput and utilization it can achieve, and the lower its contribution to the overall x-factor. 
   The machine scheduling agent  410  cannot know what pWind a consumer may choose. Thus, it cannot build the FDF into the BCF, but must pass the FDF to the consumer, which evaluates it and multiplies the BCF by the FDF in estimating the desirability of various candidate engagements. 
   The BCF, discussed above, defines the cost of processing per unit time (hourly rate) as a function of the date/time when processing occurs. The BCF is represented as an x/y table over evenly spaced time intervals. If the time interval is small enough, the BCF may be considered constant within the interval. This approach may allow the BCF math to be performed with simple matrix algebra operations. Interpolation between coarsely spaced points is not generally conducted, because the BCF may reverse its direction frequently. 
   The BCF is computed by applying the RTU function to the Committed Capacity Curve (CCC) as follows. In the following example, it is assumed that optimizing for both batching and setup occurs. If not, the unused values are omitted. The x values of the BCF table are computed at intervals of Δt at times, t i =t0+i*Δt, where i ranges from 0 to N−1 for a table with N entries, and with Δt chosen small enough that the cost at t i  is an acceptable estimate for the entire interval t i ≦t&lt;t i+1 . For notational clarity, mX and bX are represented as functions of time. The procedure for computing the BCF includes:
         1. Compute the various cc(t) values:
 
 ccSameBatch ( t   i )= mSameBatch ( t   i )* t   i   +bSameBatch (t i )  (3) 
 
 ccDiffBatch ( t   i )= mDiffBatch ( t   i )* t   i   +bDiffBatch (t i )  (4) 
 
 ccSameSetup ( t   i )= mSameSetup ( t   i )* t   i   +bSameSetup ( t   i )  (5) 
 
 ccDiffSetup ( t   i )= mDiffSetup ( t   i )* t   i   +bDiffSetup (t i )  (6) 
 
 cc ( t   i )= ccSameBatch ( t   i )+ ccDiffBatch ( t   i )  (7) 
   2. Evaluate the RTU at the appropriate cc(t) values and multiply by the urgency discount, UDF(t). This multiplication may take place either in the lot scheduling agent  405  or in the machine scheduling agent  410 . The UDF describes the state of the process tool  115 , not the lot  130 , so philosophically this computation belongs in the machine scheduling agent  410 . However, the machine scheduling agent  410  is more likely to be constrained by computational capacity than the lot scheduling agent  405 . In general, it is expected that the number of lots  130  considering using a process tool  115  at a given time will be greater than the number of process tools  115  available to a lot  130  for a given process-operation, so lots  130  will have fewer bids  435  to evaluate per bidding cycle than will process tool  1115 . Hence, the load may be better balanced if the lot scheduling agent  405  performs some computations that might typically be associated with the machine scheduling agent  410 . However, in some embodiments, the machine scheduling agent  410  may be configured to compute the UDF.
 
 BCF ( t   i )= UDF ( t )* RTU ( cc ( t   i ),  ccSameBatch ( t   i ),  ccDiffBatch (t i ),  ccSameSetup ( t   i ),  ccDiffSetup ( t   i )  (8) 
       

   To evaluate the BCF at time t, t i  is chosen such that t i ≦t&lt;t i+1 , and the value in the BCF table at time t i  is used. 
     FIG. 11  conceptually illustrates the flow for computing the BCF. The machine scheduling agent  410  passes the values for ccSame and ccDiff to the RTU function in step  1101 . In step  1102 , the machine scheduling agent  410  calculates a rate function using the RTU. In step  1103 , the machine scheduling agent  410  multiples the rate by the urgency discount function (UDF) and generates the BCF in step  1104 . Subsequently, the machine scheduling agent  410  passes the BCF to the lot scheduling agent  405  which, in turn, searches the BCF to identify candidate bids. The lot scheduling agent  405  may also apply a flexibility discount, FDF depending on the size of the commitment window it selects. 
   The bid construction phase of the machine scheduling agent  410  is now described in greater detail. The machine scheduling agent  410  receives a request bid message  425  and responds by returning the bid  435 . 
   In its bid  435 , the machine scheduling agent  410  returns a slice of the RTU function at the current ccDiff avg , as well as the portion of the BCF function that includes the time interval [EST, LDT]. This “slicing” is illustrated using the joint batch/setup scenario RTU discussed above. The normalization constants are not affected by slicing, and only the unnormalized sum is considered here. The machine scheduling agent  410  computes the constants ccDiffBatch avg  and ccDiffSetup avg  over the time period [EST, LDT]. Then the unnormalized sum becomes the following (where underlined expressions become constants because of the slicing): 
   Congestion factor:
         incV[ccSameSetup, 10, 1−ccDiffSetup avg , 100.0]       

   Batch factors:
         decV[ccSameBatch, 10, 0, 100,0];   decV[ccSameBatch, 1, 0, 50,0];   incV[ccDiffBatch avg , 1, 1, 200, 0];       

   Setup factors:
         decV[ccSameSetup, 10, 0, 100,0];   decV[ccSameSetup, 1, 0, 50,0];   incV[ccDiffSetup avg , 1, 1, 200,0].       

   If the process tool  115  is not capable of batch processing, the three batch factors are eliminated. If the process tool  115  does not require setup time to change setups, the three setup factors are eliminated. 
   The machine scheduling agent  410  computes EST by determining the expected Transport Time, TT, between the last location and the current process tool  115  and then adds this Transport Time, TT, to the Transport Start Time, TST, provided in the request bid message  425 . If the process tool  115  requires loading time, then the expected loading time, ELT, is also added
 
 EST=TST+TT+ELT.  
 
   The machine scheduling agent  410  computes the average committed capacities ccSameSetup avg  and ccDiffSetup avg  (as well as ccSameBatch avg  and ccDiffBatch avg  if it is a batching machine) within the time window [EST, LDT] using the data maintained in the list of engagements for the process tool  115 . Each average is the weighted average of the segments of the corresponding Committed Capacity Curve in the interval [EST, LDT]. The average committed capacity of each segment is weighted by the size of the time interval that defines the endpoints of the segment. For each complete segment, the average committed capacity ccX seg  for the segment defined by the interval [t s , t e ] (where X is SameBatch, DiffBatch, SameSetup, or DiffSetup) is:
 
 ccX   seg   =ccX [( t   s   +t   e )/2], 
         and the weight of the segment is:
 
 W   seg =(t e   −t   s ) 
       

   The average committed capacity, ccX avg , is computed from all whole segments between EST and LDT plus at most two partial segments if EST and/or LDT fall within the time boundaries of a segment. Assuming EST falls within a segment y=a s x+b s  bounded by [ts s ,te s ] and LDT falls within a segment y=a e x+b e  bounded by [ts e ,te e ]. The average committed capacity, ccX, and weight W of these partial segments EST and LDT are:
 
 ccX   EST =( ccX[EST]+ccX[te   s ])/2; 
 
 ccX   LDT =( ccX[ts   e   +ccX[LDT ])/2; 
 
 W   EST =( te   s   −EST ); and 
 
 W   LDT =( LDT−ts   e ). 
 
   Then, the average committed capacity, ccX avg , within the interval [EST, LDT] is:
 
 ccX   avg =( ccX   EST   *W   EST   +ccX   LDT +Σ( ccX   segi   *W   segi ))/( W   EST   +W   LDT   +ΣW   seg,i ), 
         where the summation is over all whole segments between EST and LDT.       

   The discussion now turns to the actions of the lot scheduling agent  405  and the machine scheduling agent  410  in confirming a bid  435 . When a lot scheduling agent  405  awards a bid  435  to a process tool  115 , the machine scheduling agent  410  confirms the award if the lot  130  can afford the actual cost of the bid  435  and the actual cost has not increased by more than a configurable percentage of the original cost. The engagement returned by the lot scheduling agent  405  to the machine scheduling agent  410  for bid confirmation defines a piecewise linear function of time E(t), as outlined in further detail below. 
   The machine scheduling agent  410  calculates the actual cost of the engagement using the following steps. In recalculating various functions, the machine scheduling agent  410  must cache the previous values and be able to restore them in case the bid  435  is not confirmed.
         a) Recalculate the urgency discount as outlined above in Equation 1 using the cc(t) function in effect at the time the lot  130  awards the bid  435  to the process tool  115 . Note that this may be a different cc(t) function than the one used in computing the urgency discount originally offered to the lot, since if the lot has been sluggish in responding, other lots  130  may have filled up the short-term lack of work. However, this engagement is not included in the cc(t) function used in computing the urgency discount. Thus, this step must precede the following one.   b) Recalculate the ccSameX(t), ccdiff(X(t), and cc(t) functions including the new engagement, as defined in Equations 3-7 above. Note that ccDiffX(t), where X=batch or setup, does not change as a result of the lot  130 , and therefore, it may not require recalculation.   c) Calculate a new BCF as defined in Equation 8 above by evaluating the RTU function with the new committed capacity values calculated in step b.   d) Calculate the flexibility discount FD on the basis of the new engagement, as defined by Equation 8 above.   e) Compute the total cost of the engagement:
 
 C=FD*Σ ( BCF ( t )* E ( t )), 
   where the summation is over all entries in the BCF table such that t s ≦t&lt;t e .       

   The machine scheduling agent  410  denies the award if the lot  130  does not have sufficient budget to afford it or if the actual cost has increased more than a configurable percentage of the original cost. 
   Another important feature of the machine scheduling agent  410  is to monitor the committed capacity of the process tool  115  to determine if it has overcommitted its resources. Each time a change occurs to the committed capacity of the process tool  115 , the machine scheduling agent  410  runs a background task that looks for regions of violation (ROVs) where the committed capacity curve exceeds the maximum capacity of the machine, and generates candidate moves of selected engagements to try to reduce or eliminate each ROV. 
   The “maximum capacity” of process tool  115  i is defined as:
 
 MaxCap   i   =MaxLots   i   −SetupAllowance   i   −SafetyAllowance   i , where 
         MaxLots is the maximum number of lots  130  that the process tool  115  can process concurrently. For non-batching process tools  115 , MaxLots=1. For batching process tools  115 , MaxLots is the size of a full batch.   SetupAllowance adjusts for the fact that the kernels for individual lots  130  do not include setup (because an individual lot  130  cannot know whether or not it will need a new setup). Computation of the SetupAllowance is discussed below.   SafetyAllowance is a tuning factor. The Committed Capacity curve is a heuristic estimate of the availability of the process tool  115 . Sometimes a process tool  115  may not be able to process a lot  130  even in a period where there is no ROV. Increasing the SafetyAllowance reduces the likelihood of such an event. Conversely, a negative SafetyAllowance permits a process tool  115  to overbook intentionally. Initially, SafetyAllowance should be set to a value reflecting the historical level of unplanned downtime on the resource.       

   To compute the SetupAllowance, observe that a setup can be viewed as a task that completely consumes the capacity of the process tool  115  (MaxLots) for the duration of the setup. If, on average, the process tool  115  spends s % of its time in setup operations, its overall capacity is reduced by:
 
 SetupAllowance=s*MaxLots /100. 
 
   Each process tool  115  maintains a Exponentially Weighted Moving Average (EWMA) of the percentage of its time it spends in setup, and uses the latest value of this average to compute MaxCap each time it scans its schedule of engagements for ROVs. For example, if the ith estimate of s is s i  and the resource has spent x % of its time in setup since s i  was calculated, the next estimate of s is given by:
 
 s   i+1 =(1−λ) s   i   +λx =0.8 s   i +0.2 x,  
 
where Lambda is set at 0.2 in the illustrated embodiment. Of course other values of Lambda may be used for the EWMA.
 
   The process the machine scheduling agent  410  uses for selecting candidate moves to reduce an ROV includes the following steps:
         a) Find a region where the CCC exceeds MaxCap.   b) Calculate the ROV Area, the area of the CCC above MaxCap.   c) Identify all engagements with working windows that overlap the region of violation, and compute their contribution to the ROV. The contribution of engagement i to the ROV is ROV i .   d) For each engagement i with an overlapping working window, determine zero or more moves the engagement could make to reduce ROV i . An engagement can provide more than one move. Moves can be:
           shift left (move the entire WW left without changing its size);   shift right (move the entire WW right without changing its size);   shrink left (shrink the WW by moving its right end to the left); and   shrink right (shrink the WW by moving its left end to the right).   A move to expand left or right is typically not permitted in order to avoid oscillation. The following heuristics should be used for moves depending on the way the working window (WW) overlaps with the ROV. There are four cases:   
           Case 1: WW overlaps from left side of ROV.
           Options are shift or shrink left.   
           Case 2: WW overlaps from right side of ROV.
           Options are shift or shrink right.   
           Case 3: WW overlaps entirely within ROV.
           Options are shift left or right.   
           Case 4: WW overlaps from both sides of ROV.
           Options are shift left, shrink left, shrink right, or shift right.   
               

   An exemplary technique for computing the amount of shift or shrinkage necessary on a given engagement to release a specified amount of ROV area is discussed in greater detail below. In general, the technique includes, for each engagement i, computing each candidate move j to provide the maximum area ΔA i,j , up to the total contribution available from engagement i ROV i , allowed within the constraints of engagement i&#39;s commitment window.
         e) Compute the change in cost for each move i considered. To compute the change in cost:
           Compute the base C 1 , the cost of the unmodified engagement using its BCF and the flexibility discount, FDF, appropriate to its pWind. This cost is the same for all candidate moves generated from a single engagement.   Compute the new BCF that would be in effect for the moved engagement. This computation involves computing the new ccX curves, evaluating the RTU, and applying the urgency discount.   Compute C 2i , the cost of the new engagement generated by move i, using the new BCF and the flexibility discount appropriate to the new pWind.   The change in cost is ΔC i =C 2i −C 1 .   The machine scheduling agent  410  chooses the one move that provides the minimum cost per area resolved ΔC i /ΔA i  and executes that move.   
               

   This ends an iteration of ROV resolution. The machine scheduling agent  410  may decide that it is not feasible to resolve an ROV. A configurable control MinROVpercent defines the minimum percentage of the ROV area that a candidate move must return. If the process tool  115  has no move options, or if the best move option returns less than MinROVpercent of the ROV area, the process tool  115  gives up on resolving the ROV and cancels one of the engagements contributing to the ROV, paying the penalty refund described above. The engagement cancelled should be the engagement with the lowest priority, which means the lot  130  with the minimum “normalized” budget, i.e., budget per unit of processing (kernel) time. 
   The technique for resolving ROVs is now described in greater detail in reference to  FIG. 12 , which shows an exemplary engagement density curve. In general, the engagement density curve is a trapezoid (i.e., although in a special case, it may be a triangle), with the distinguishing parameters shown in FIG.  12 . Three items constitute the definition of an engagement:
         ws: window start   we: window end   k: kernel width (not visible in FIG.  12 ).       

   Several other parameters can be derived from these:
         s: slack=we−ws−k (excess room in the window). The slack, s, is a critical factor in the overall shape of the curve. When s=k, the engagement is a triangle, otherwise it is a trapezoid, but the parameters of the trapezoid differ depending on whether s&lt;k or s&gt;k.   p: plateau height=Min(k/s, 1). When s&lt;k, p=1, otherwise p=k/s.   ps: plateau start=Min(ws+k, we−k). When s&lt;k, ps=we−k, otherwise ps=ws+k.   pe: plateau end=Max(ws+k, we−k). When s&lt;k, pe=ws+k, otherwise pe=we−k.       

   Note that s*p=Min(s, k), as this will be used in simplifying equations below. 
     FIG. 12  also identifies three possible time intervals:
         ws&lt;t 1 &lt;ps   ps&lt;t 2 &lt;pe   pe&lt;t 3 &lt;we       
   The curve is constructed so that the total area under it is k. When it partially overlaps an ROV, we need to compute the area it contributes to the overlap. This computation amounts to evaluating the area between t and one end or the other of the engagement, where t is the edge of the ROV. Without loss of generality, the case where the ROV is to the right of t, so that the area required is between t and we is considered.  FIG. 13  illustrates a prototypical situation with an ROV. The curve  1310  is the overall cc(t) curve, the horizontal and vertical lines show where cc(t)&gt;1 at t=5.76, and the trapezoid  1320  is one of the engagements that make up cc(t), with ws=1, we=10, and k=2. 
   To determine the amount of overlap between the trapezoid  1320  and the ROV, a function to compute the area extending into the ROV is developed.
 
RightArea(t, ws, we, k). 
 
   All of the functions defined below are functions of (t, ws, we, k), but at this point only the dependence on t is examined. For clarity and ease of illustration the last three arguments are omitted. 
   The area of the curve can be divided into the plateau region and the two ramps. The ramps are symmetrical. Each has the area:
 
 RampArea=p *( ps−ws )/2. 
 
   The second factor may be simplified as:
 
( ps−ws )=Min( ws+k, we−k )− ws =Min( k, we−ws−k )=Min( k, s )= s*p.  
 
   Substituting this result into the RampArea equation yields:
 
 RampArea =Min( k/s , 1)*Min( k, s )/2 =s*p   2 /2. 
 
   The central plateau has the area:
 
 PlateauArea=p* ( pe−ps ). 
 
   Again, the equation may be simplified as:
 
( pe−ps )=Max( ws+k, we−k )−Min( ws+k, we−k )=|( ws+k )−( we−k )|=| k −( we−ws−k )|=| k−s|,  
 
resulting in:
 
 PlateauArea =Min( k/s,  1)*| k−s|=p*|k−s|.  
 
   Note that when s&lt;k, |k−s|=k−s, otherwise |k−s|=s−k. 
   Partial overlap regions may occur if t falls in the middle of a ramp or of a plateau. If t occurs somewhere in the middle of a ramp, the ramp is divided into two parts. Consider a left-hand ramp  1400 , between ws and ps, as shown in FIG.  14 . The ramp  1400  is linear, so for ws&lt;t&lt;ps, the height of the ramp at t is just the total height multiplied by the proportion of the distance that t is from ws to ps.
 
 LeftRampHeight ( t )= p *( t−ws )/( ps−ws )=( t−ws )/ s  
 
   The ramp  1400 , the vertical line at t, and the horizontal line at LeftRampHeight divide the box p*(ps−ws) into six lettered regions a-e. The area to the left of t, area c, is referred to as the LeftRampLeftArea(t), and is defined by:
 
 LeftRampLeftArea ( t )= LeftRampHeight ( t )*( t−ws )/2=( t−ws)   2 /(2 *s ). 
 
   The area of the remaining portion of the ramp  1400 , areas d and C may be calculated by:
 
 LeftRampRightArea ( t )= RampArea ( t )− LeftRampLeftArea ( t )= s*p   2 /2−( t−ws ) 2 /(2 *s ). 
 
   By symmetry, the right ramp parameters are:
 
 RightRampHeight ( t )=( we−t )/ s  
 
 RightRampRightArea ( t )= RightRampHeight ( t )*( we−t )/2=( we−t ) 2 /(2 s ), and 
 
 RightRampLeftArea ( t )= sp   2 /2−( we−t ) 2 /(2 s ) 
 
   The other partial overlap situation occurs when t is in the middle of the plateau. This case is simpler, since the plateau height is a constant. The partial plateau area is computed by:
 
 PlateauLeftArea ( t )= PlateauArea *( t−ps )/( pe−ps )=Min( k/s , 1)*| k−s |*( t−ps )/| k−s |=Min( k/s, 1)*( t−ps )= p *( t−ps ); and 
 
 PlateauRightArea ( t )= PlateauArea *( pe−t )/( pe−ps )=Min( k/s,  1)*( pe−t )= p *( pe−t ). 
 
   As  FIG. 12  shows, the shape of the curve changes discontinuously at ps and pe. Four cases may be distinguished, depending on where t falls on the engagement being evaluated. In each case, the overall area to the right or left of t may be determined either by adding together a combination of whole and partial regions, or by subtracting a partial region from the known area k of the whole trapezoid. 
   Case 1: ws&lt;t&lt;ps:
 
 RightArea 1( t )= k−LeftRampLeftArea ( t )= k− ( t−ws ) 2 /(2 *s ) 
 
 LeftArea 1( t )= LeftRampLeftArea ( t )=( t−ws ) 2 /(2 *s ) 
 
Case 2: ps&lt;t&lt;pe:
 
 RightArea 2( t )= PlateauRightArea ( t )+ RampArea=p *( pe−t )+ s*p   2 /2 
 
 LeftArea 2( t )= RampArea+PlateauLeftArea ( t )= s*p   2 /2 +p *( t−ps ) 
 
Case 3: pe&lt;t&lt;we:
 
 RightArea 3( t )= RightRampRightArea ( t )=( we−t ) 2 /(2 s ) 
 
 LeftArea 3( t )=k− RightRampRightArea ( t )= k −( we−t ) 2 /(2 s ) 
 
Case 4: t&lt;ws (ww Overlaps Entirely within ROV):
 
 RightArea 4( t )= k  
 
LeftArea4( t )=0 
 
   The situation shown in  FIG. 13  falls in Case 2, and RightArea2(t=5.76, 1, 10, 2)=0.93. 
   There are two mechanisms for reducing the area that an engagement contributes to cc(t): shifting the whole engagement (presumably because the working window is smaller than the commitment window) and shrinking the working window. In the example scenario of  FIG. 13 , where the engagement is on the left end of the ROV, the engagement may be shifted left or shrunk by moving its right end to the left. The machine scheduling agent  410  calculates how much to shift or shrink the working window to realize a required amount of ROV reduction. 
   The excess area in the ROV is shown in  FIG. 13  (the area under cc(t) but above cc(t)=1) is 6.24. By shifting the engagement represented by the trapezoid  1420  entirely out of the ROV area, 0.93 could be recovered as indicated above. However, in general, it may be desirable to take only a portion of the available area reduction. Up to now the functions have been in the form of area=f(t). To determine desired area recoveries, functions of the form t=g(area) are developed. 
     FIG. 15  shows the effect of shifting the engagement to the left by Δt. The upper picture shows the initial position of the engagement. The CCC (not shown) begins to exceed the threshold at time t, and the ROV is to the right of t. The total contribution of the engagement to the ROV is the shaded portion  1500  to the right of t. The lower picture shows the engagement after being shifted at to the left. Now the only contribution of the engagement to the ROV is the small triangle  1510  that remains to the right of t. The area that has been removed from the ROV is the shaded region  1520  between t and t−Δt. The same area on the original trapezoid is delimited by t and t+Δt. Thus, the saved area on the original engagement can be computed as:
   SavedArea=RightArea ( t )− RightArea ( t+Δt ),  
leaving the parameters that describe the engagement (ws, we, k) unchanged.
 
     FIG. 16  shows the effect of shifting the engagement to the right by Δt. The upper picture shows the initial position of the engagement. The CCC (not shown) begins to exceed the threshold at time t, and the ROV is to the left of t. The total contribution of the engagement to the ROV is the shaded portion  1600  to the left of t. The lower picture shows the engagement after being shifted Δt to the right. Now the only contribution of the engagement to the ROV is the small triangle  1610  that remains to the left of t. The area that has been removed from the ROV is the shaded region  1620  between t and t+Δt. The same area on the original trapezoid is delimited by t and t−Δt. Thus the saved area on the original engagement can be computed as:
   SavedArea=LeftArea ( t )− LeftArea ( t−Δt ),  
leaving the parameters that describe the engagement (ws, we, k) unchanged.
 
   The time difference, Δt, required to move an engagement depends on two factors: which case t satisfies, and how much area is needed, AreaNeeded. Of course, Δt time may not be available to shift, but for the moment this case is ignored to simplify the discussion here. The discussion below presents exemplary cases for each equation. These cases are based on the engagement (ws,we,k)=(1,10,2), whose plateau begins at 3 and ends at 8. Total area available is equal to the kernel, 2. 
   Case 1: ws&lt;t&lt;ps (ROV to Right). 
   An engagement that overlaps an ROV to the right of it has three components that may contribute to the ROV: LeftRampRightArea(t), PlateauArea, and RampArea. Depending on the AreaNeeded, part or all of these components may be required. For purposes of the following example, let t=2.
 
Subcase 1.1: If AreaNeeded&lt;LeftRampRightArea(t), then both t and t+Δt fall in the left ramp, and Δt must satisfy:
 
             AreaNeeded   =       ⁢       RightArea1   ⁢           ⁢     (   t   )       -     RightArea1   ⁢           ⁢     (     t   +     Δ   ⁢           ⁢   t       )                     =       ⁢     k   -         (     t   -   ws     )     2     /     (     2   *   s     )       -   k   +         (     t   +     Δ   ⁢           ⁢   t     -   ws     )     2     /     (     2   *   s     )                     =       ⁢       [         (     t   +     Δ   ⁢           ⁢   t     -   ws     )     2     -       (     t   -   ws     )     2       ]     /     (     2   *   s     )                   =       ⁢     Δ   ⁢           ⁢   t   ⁢           ⁢       (       Δ   ⁢           ⁢   t     +     2   ⁢           ⁢     (     t   -   ws     )         )     /       (     2   ⁢           ⁢   s     )     .                   
 
   This formula is quadratic with respect to Δt, and so yields two solutions. The solution that yields Δt&gt;0 is:
 
Δ t=ws−t+√{square root over (2·AreaNeeded·s+(t−ws)     2     )}.  
         Example: A shift of 0.5 yields area of 0.089.
 
Subcase 1.2: If LeftRampRightArea(t)&lt;AreaNeeded&lt;LeftRampRightArea(t)+PlateauArea, then t+Δt falls in the plateau, and Δt must satisfy:
 
 AreaNeeded=LeftRampRightArea ( t )+ PlateauLeftArea ( t+Δt )=s*p 2 /2−( t−ws ) 2 /(2 *s )+ p *( t+Δt−ps ), 
 
which can be solved to yield:
 
Δ t=AreaNeeded/p−t+ps−s*p/ 2+( t−ws)   2 /(2 s*p ). 
   Example: A shift of 2 yields area of 0.5.
 
Subcase 1.3: If LeftRampRightArea(t)+PlateauArea&lt;AreaNeeded&lt;LeftRampRightArea(t)+PlateauArea+RampArea, then t+Δt falls in the right ramp, and Δt must satisfy:
 
             AreaNeeded   =       ⁢       LeftRampRightArea   ⁢           ⁢     (   t   )       +   PlateauArea   +                     ⁢     RightRampLeftArea   ⁢           ⁢     (     t   +     Δ   ⁢           ⁢   t       )                   =       ⁢       s   *     p   2     ⁢   2     -         (     t   -   ws     )     2     /     (     2   *   s     )       +     p   ⁢           ⁢          k   -   s            +     s   *       p   2     /   2       -                     ⁢         (     we   -   t   -     Δ   ⁢           ⁢   t       )     2     /     (     2   *   s     )                   =       ⁢       s   ·     p   2       +     p   ·          k   -   s            -             (     t   -   ws     )     2     +       (     we   -   t   -     Δ   ⁢           ⁢   t       )     2         2   ·   s       .                 
 
Again, the equation is quadratic in Δt, and both solutions yield positive Δt, but the valid solution is: 
         Δ   ⁢           ⁢   t     =     we   -   t   -             -   2     ·   AreaNeeded   ·   s     +     2   ⁢     p   ·   s   ·          k   -   s              +     2   ⁢     p   2     ⁢           ⁢     s   2       -       (     t   -   ws     )     2         .           
   Example: A shift of 7 yields area of 1.86.
 
Subcase 1.4: If LeftRampRightArea(t)+PlateauArea+RampArea&lt;AreaNeeded, the engagement must be moved completely out of the ROV, so Δt=we−t, and other sources must be found if additional area is needed.
       

   Case 1′: ws&lt;t&lt;ps (ROV to Left) 
   An engagement that overlaps an ROV to the left of it contributes at most LeftRampLeftArea(t), yielding two subcases. 
   Subcase 1′.1: If AreaNeeded&lt;LeftRampLeftArea(t), then t−Δt falls in the left ramp, and Δt must satisfy:
 
 AreaNeeded=LeftRampLeftArea ( t )− LeftRampLeftArea ( t−Δt )=Min( k/s,  1)*(( t−ws ) 2 −( t−Δt−ws ) 2 )/(2*Min( k,s ))=Min( k/s,  1)*(Δ t *(2 *t− 2 *ws−Δt ))/(2*Min( k, s )), 
 
which can be expressed as: 
         AreaNeeded   =         (     t   -   ws     )     ·       Δ   ⁢           ⁢   t     s       -       Δ   ⁢           ⁢     t   2         2   ·   s           ,       
 
a quadratic that can be solved as: 
         Δ   ⁢           ⁢   t     =     t   -   ws   -         Min   ⁢           ⁢       (       k   /   s     ,   1     )     ·     [       Min   ⁢           ⁢       (       k   /   s     ,   1     )     ·       (     t   -   ws     )     2         -     2   ·   AreaNeeded   ·     Min   ⁡     (     k   ,   s     )           ]             Min   ⁢           ⁢     (       k   /   s     ,   1     )               
     or     
         Δ   ⁢           ⁢   t     =     t   -   ws   -             (     t   -   ws     )     2     -     2   ·   s   ·   AreaNeeded         .           
 
Subcase 1′.2: If LeftRampLeftArea(t)&lt;AreaNeeded, then the engagement must be moved completely out of the ROV, so Δt=t−ws, and other sources must be found if additional area is needed.
 
Case 2: ps&lt;t&lt;pe (ROV to Right).
 
   An engagement that overlaps an ROV to the right of it contributes at most PlateauRightArea(t)+RampArea, yielding three subcases. For purposes of this illustration, let t=4. 
   Subcase 2.1: If AreaNeeded&lt;PlateauRightArea(t), then t+αt falls in the plateau, and Δt must satisfy:
 
 AreaNeeded=PlateauRightArea ( t )− PlateauRightArea ( t+Δt )= p ( pe−t )− p ( pe−t−Δt )= p Δt,  
 
yielding:
 
Δ t=AreaNeeded/p.  
         Example: A shift of 2 yields area of 0.57.
 
Subcase 2.2: If PlateauRightArea(t)&lt;AreaNeeded&lt;PlateauRightArea(t)+RampArea, then t+Δt falls in the right ramp, and Δt must satisfy:
 
 AreaNeeded=PlateauRightArea ( t )+ RightRampLeftArea ( t+Δt )= p ( pe−t )+ s*p   2 /2−( we−t−Δt)   2 /(2 s ). 
 
The valid solution to the quadratic is: 
         Δ   ⁢           ⁢   t     =     we   -   t   -             -   2     ·   s   ·   AreaNeeded     +     2   ⁢     p   ·   s   ·   pe       +       p   2     ⁢           ⁢     s   2       -     2   ⁢     p   ·   s   ·   t           .           
   Example: A shift of 5 yields area of 1.36.
 
Subcase 2.3: If PlateauRightArea(t)+RampArea&lt;AreaNeeded, then the engagement must be moved completely out of the ROV, so Δt=we−t, and other sources must be found if additional area is needed.
 
Case 2′: ps&lt;t&lt;pe (ROV to Left).
       

   An engagement that overlaps an ROV to the left of it contributes at most PlateauLeftArea(t)+RampArea, yielding three subcases. 
   Subcase 2′.1: If AreaNeeded&lt;PlateauLeftArea(t), then t−Δt falls in the plateau, and Δt must satisfy:
 
 AreaNeeded=PlateauLeftArea ( t )− PlateauLeftArea ( t−Δt )=Min( k/s,  1)*(( t−ps )−( t−Δt−ps ))=Δ t *Min( k/s, 1), 
 
yielding:
 
Δ t=AreaNeeded /Min( k/s,  1)= AreaNeeded/p.  
 
Subcase 2′.2: If PlateauLeftArea(t)&lt;AreaNeeded&lt;PlateauLeftArea(t)+RampArea, then t−Δt falls in the left ramp, and Δt must satisfy:
 
 AreaNeeded=PlateauLeftArea ( t )+ LeftRampRightArea ( t−Δt )=Min( k/s,  1)*( t−ps )+( ps−t+Δt )*( p +Min( k/s , 1)*( t−Δt−ws )/Min( k,s ))/2, 
 
which can be expressed as: 
       AreaNeeded   =       p   ·     (     t   -     p   ⁢           ⁢   s       )       +       (       p   ⁢           ⁢   s     -   t   +     Δ   ⁢           ⁢   t       )     ·           p   ·   s     +   t   -     Δ   ⁢           ⁢   t     -   ws       2   ·   s       .             
 
   The quadratic equation of Δt, can be rewritten as:
 
Δt 2 −( p*s +2 *t−ps−ws )*Δt−( t−ps )*( p*s−t+ws )+2 *s*AreaNeeded =0, 
 
and solved for Δt to yield: 
           Δ   ⁢           ⁢   t     =       a   -     b       2       ,       
         where   a=p·s+2·t−ps−ws, and   b=(p·s+2·t−ps−ws) 2 −8·AreaNeeded+4·(t−ps)·(p·s−t+ws).
 
Subcase 2′.3: If PlateauLeftArea(t)+RampArea&lt;AreaNeeded, then the engagement must be moved completely out of the ROV, so Δt=t−ws, and other sources must be found if additional area is needed.
 
Case 3: pe&lt;t&lt;we (ROV to Right)
 
An engagement that overlaps an ROV to the right of it contributes at most RightRampRightArea(t), yielding two subcases. For purposes of this illustration, let t=8.5.
 
Subcase 3.1: If AreaNeeded&lt;RightRampRightArea(t), then t+Δt falls in the right ramp, and Δt must satisfy:
 
 AreaNeeded=RightRampRightArea ( t )− RightRampRightArea ( t+Δt )=(( we−t ) 2 −( we−t−Δt ) 2 )/(2 s ), 
 
which is a quadratic equation of Δt that can be solved to yield: 
         Δ   ⁢           ⁢   t     =     we   -   t   -             (     we   -   t     )     2     -     2   ·   s   ·   AreaNeeded         .           
   Example: A shift of 1 yields area of 0.14.
 
Subcase 3.2: If RightRamptRightArea(t)&lt;AreaNeeded, then the engagement must be moved completely out of the ROV, so Δt=we−t, and other sources must be found if additional area is needed.
 
Case 3′: pe&lt;t&lt;we (ROV to Left).
       

   An engagement that overlaps an ROV to the left of it may contribute three components to the ROV: RightRampLeftArea(t), PlateauArea, and RampArea. Depending on AreaNeeded, part or all of these components may be required. 
   Subcase 3′. 1: If AreaNeeded&lt;RightRampLeftArea(t), then t−Δt falls in the right ramp, and Δt must satisfy:
 
 AreaNeeded=RightRampLeftArea ( t )− RightRampLeftArea ( t−Δt )=( t−pe )*Min( k/s,  1)*(1+( we−t )/Min( k,s ))/2−( t−Δt−pe )*Min( k/s,  1)*(1+( we−t+Δt )/Min( k/s ))/2, 
 
which can be rewritten as: 
       AreaNeeded   =           t   -   pe     2     ·     (     p   +       we   -   t     s       )       -         t   -     Δ   ⁢           ⁢   t     -   pe     2     ·       (     p   +       we   -   t   +     Δ   ⁢           ⁢   t       s       )     .             
 
   This quadratic may be simplified to:
 
Δ t   2 +( s·p+pe+we− 2· t )·Δ t− 2 ·s·AreaNeeded= 0, 
 
and solved for Δt to yield: 
         Δ   ⁢           ⁢   t     =         -     (       s   ·   p     +   pe   +   we   -     2   ·   t       )       +           (       s   ·   p     +   pe   +   we   -     2   ·   t       )     2     +     8   ·   s   ·   AreaNeeded           2         
 
Subcase 3′.2: If RightRampLeftArea(t)&lt;AreaNeeded&lt;RightRampLeftArea(t)+PlateauArea, then t−Δt falls in the plateau, and Δt must satisfy:
 
 AreaNeeded=RightRampLeftArea ( t )+ PlateauLeftArea ( t−Δt )=( t−pe )*Min( k/s,  1)*(1+( we−t )/Min( k,s ))/2+Min( k/s,  1)*( pe−t+Δt )=Min( k/s,  1)*(( t−pe )*(1+( we−t )/Min( k/s ))/2+( pe−t+Δt )), 
 
which can be rewritten as: 
         AreaNeeded   =           t   -   pe     2     ·     (     p   +       we   -   t     s       )       +     p   ·     (     pe   -   t   +     Δ   ⁢           ⁢   t       )           ,       
 
and solved to yield:
 
Δ t =( t−pe )*( t−we )/(2*Min( k,s ))−( pe−t )/2 +AreaNeeded /Min( k/s , 1) 
 
or 
         Δ   ⁢           ⁢   t     =           (     t   -   pe     )     ·     (       p   ·   s     +   t   -   we     )         2   ·   s   ·   p       +       AreaNeeded   p     .           
 
   Subcase 3′.3: If RightRampLeftArea(t)+PlateauArea&lt;AreaNeeded&lt;RightRampLeftArea(t)+PlateauArea+LeftRampArea, then t−Δt falls in the left ramp, and Δt must satisfy:
 
 AreaNeeded=RightRampLeftArea ( t )+ PlateauArea+LeftRampRightArea ( t−Δt )=( t−pe )*Min( k/s,  1)*(1+( we−t )/Min( k,s ))/2+Min( k/s , 1)*|k−s|+( ps−t+Δt )*( p +Min( k/s,  1)*( t−Δt−ws )/Min( k,s ))/2, 
 
which can be expressed as: 
       AreaNeeded   =         (     t   -   pe     )     ·         p   ·   s     +   we   -   t       2   ·   s         +     p   ·          k   -   s            +       (     ps   -   t   +     Δ   ⁢           ⁢   t       )     ·         p   ·   s     +   t   -     Δ   ⁢           ⁢   t     -   ws       2   ·   s               
 
and simplified to yield:
 
Δ t   2 −( s·p+ 2 ·t−ps−ws )−Δ t+ 2 −s·AreaNeeded +( t−ps )·( s·p+t−ws )−2 ·s·p·|k−s|− ( t−pe )−( s·p+we−t )=0 
 
Solving for Δt yields: 
         Δ   ⁢           ⁢   t     =       a   2     -                 a   2     -     8   ·   s   ·   AreaNeeded     -     4   ·     [         (     t   -   ps     )     ·     (       s   ·   p     +   t   -   ws     )       -                         2   ·   s   ·   p   ·          k   -   s            -       (     t   -   pe     )     ·     (       p   ·   s     +   we   -   t     )         ]             2           
 
where
         a=(s·p+2·t−p−ws).
 
Subcase 3′.4: If RightRampLeftArea(t)+PlateauArea+RampArea&lt;AreaNeeded, then the engagement must be moved completely out of the ROV, so Δt=t−ws, and other sources must be found if additional area is needed.
 
Case 4: t&lt;ws (WW Overlaps Entirely within ROV and Shift Engagement Left).
       

   An engagement that entirely overlaps an ROV has three components that may contribute to the ROV: LeftRampArea, PlateauArea and RightRampArea. Depending on the AreaNeeded, part or all of these components may be required. The total shift Δt in this case has two parts: Δt 1 =ws−t (which does not reduce any ROV) and Δt 2 =t−ws′ (which is the special case of Case 1 when t=ws), where ws′ is the new window start after the shift. Thus there are four subcases. 
   Subcase 4.1: If AreaNeeded&lt;LeftRampArea, then Δt 2  can be derived directly from the subcase 1.1 with t=ws, and that yields:
 
Δ t   2 =√{square root over (2 *AreaNeeded*s )}. 
 
Thus,
 
Δ t=Δt   1   Δt   2   =ws−t+√{square root over (2*AreaNeeded*s)}.  
 
   Subcase 4.2: If LeftRampArea&lt;AreaNeeded&lt;LeftRampArea+PlateauArea, then Δt 2  can be derived directly from the subcase 1.2 with t=ws, and that yields: 
         Δ   ⁢           ⁢     t   2       =       AreaNeeded   p     -   t   +   ps   -         s   *   p     2     .           
 
Thus, the total shift is 
         Δ   ⁢           ⁢   t     =         Δ   ⁢           ⁢     t   1       +     Δ   ⁢           ⁢     t   2         =     ws   -   t   +     AreaNeeded   p     -   t   +   ps   -         s   *   p     2     .             
 
   Subcase 4.3: If LeftRampArea+PlateauArea&lt;AreaNeeded&lt;LeftRampArea+PlateauArea+RightRampArea, then Δt 2  can be derived from the subcases 1.3 with t=ws, and that yields: 
         Δ   ⁢           ⁢     t   2       =     we   -   ws   -           2   *   p   *   s   *          k   -   s            +     2   *     p   2     *     s   2       -     2   *   s   *   AreaNeeded         .           
 
Thus, the total shift is: 
         Δ   ⁢           ⁢   t     =         Δ   ⁢           ⁢     t   1       +     Δ   ⁢           ⁢     t   2         =     we   -   t   -           2   *   p   *   s   *          k   -   s            +     2   *     p   2     *     s   2       -     2   *   s   *   AreaNeeded         .             
 
   Subcase 4.4: If LeftRampArea+PlateauArea+RightRampArea&lt;AreaNeeded, the engagement must be moved completely out of the ROV, so Δt=we−t, and other sources must be found if additional area is needed. 
   Case 4′: we&lt;t (WW overlaps entirely within ROV and shift engagement right). 
   Similar to the case 4, an engagement that entirely overlaps an ROV has three components that may contribute to the ROV: LeftRampArea, PlateauArea and RightRampArea. Depending on the AreaNeeded, part or all of these components may be required. The total shift Δt in this case has two parts: Δt 1 =t−we (which does not reduce any ROV), and Δt 2 =we′−t (which is the special case of Case 3′ when t=we), where we′ is the new window end after the shift. Thus, there are four subcases. 
   Subcase 4′. 1: If AreaNeeded&lt;RightRampArea, then Δt 2  can be derived directly from the Subcase 3′.1 with t=we, and that yields: 
         Δ   ⁢           ⁢     t   2       =           -     (       s   *   p     +   pe   -   t     )       +           (       s   *   p     +   pe   -   t     )     2     +     8   *   s   *   AreaNeeded           2     .         
 
Thus, the total shift is given by: 
         Δ   ⁢           ⁢   t     =         Δ   ⁢           ⁢     t   1       +     Δ   ⁢           ⁢     t   2         =     t   -   we   +           -     (       s   *   p     +   pe   -   t     )       +           (       s   *   p     +   pe   -   t     )     2     +     8   *   s   *   AreaNeeded           2     .             
 
   Subcase 4′2: If RightRampArea&lt;AreaNeeded&lt;RightRampArea+PlateauArea, then Δt 2  can be derived directly from the Subcase 3′.2 with t=we, and that yields: 
         Δ   ⁢           ⁢     t   2       =           (     t   -   pe     )     *   p   *   s       2   *   s   *   p       +       AreaNeeded   p     .           
 
Thus, the total shift is given by: 
         Δ   ⁢           ⁢   t     =         Δ   ⁢           ⁢     t   1       +     Δ   ⁢           ⁢     t   2         =     t   -   we   +         (     t   -   pe     )     *   p   *   s       2   *   s   *   p       +       AreaNeeded   p     .             
 
   Subcase 4′3: If RightRampArea+PlateauArea&lt;AreaNeeded&lt;LeftRampArea+PlateauArea+RightRampArea, then Δt 2  can be derived directly from the Subcase 3′.3 with t=we, and that yields: 
         Δ   ⁢           ⁢     t   2       =       A   2     -                 A   2     -     8   *   s   *   AreaNeeded     ⁢           -     4   *     [         (     t   -   ps     )     *     (       s   *   p     +   t   -   ws     )       -                         2   *   s   *   p   *          k   -   s            -       (     t   -   pe     )     *   p   *   s       ]             2           
 
where A=s*p+2*t−ps−ws.
 
Thus, the total shift is then: 
         Δ   ⁢           ⁢   t     =         Δ   ⁢           ⁢     t   1       +     Δ   ⁢           ⁢     t   2         =     t   -   we   +     A   2     -                 A   2     -     8   *   s   *   AreaNeeded     ⁢           -     4   *     [         (     t   -   ps     )     *     (       s   *   p     +   t   -   ws     )       -                         2   *   s   *   p   *          k   -   s            -       (     t   -   pe     )     *   p   *   s       ]             2             
 
   Subcase 4′4: If LeftRampArea+PlateauArea+RightRampArea&lt;AreaNeeded, then the engagement must be moved completely out of the ROV, so Δt=t−ws, and other sources must be found if additional area is needed. 
   The other option for reducing the ROV is to shrink an engagement. When shrunk, the engagement density changes shape, and computation of the needed area can no longer be restricted to the original engagement. The computation is illustrated by considering the case of a shrink left. Analgous equations define the behavior of a shrink right. Instead of computing RightArea(t)−RightArea(t−Δt) as described for shift left, the required computation for shrink left is RightArea(t, ws, we, k)−RightArea(t, ws, we−Δt, k), as illustrated in FIG.  17 . 
   In the new (shrunk) engagement, relative to the original engagement,
         1. ws remains unchanged;   2. we is decreased by Δt;   3. as long as p&lt;=1(s&gt;=k), ps is constant at ws+k, and pe decreases monotonically, pe′=we−Δt−k,   4. as long as p=1(s&lt;=k), pe is constant at ws+k, and ps decreases monotonically, ps′=we−Δt−k.       

   In discussing the dynamics as Δt increases, it is sometimes useful to view t as advancing to the right, passing successive landmarks (ps, pe, we), although, in fact, these landmarks are moving toward t as the engagement shrinks. 
   An engagement has three regions, ordered from left to right as &lt;LeftRamp, Plateau, RightRamp&gt;. As a consequence of the previous four observations, for a shrink left, if t falls in one of these regions on the original engagement, it can only fall in the same region or a region more to the right on the shrunken engagement. For a shrink right, t can only fall in the same region or a region more to the left on the shrunken engagement. 
   The equation needed to solve for Δt (for a shrink left) is:
 
 AreaNeeded=RightArea ( t, ws, we, k )− RightArea ( t, ws, we−Δt, k ). 
 
   Similarly, for a shrink right, the equation needed to solve for Δt is:
 
 AreaNeeded=LeftArea ( t, ws, we, k )− LeftArea ( t, ws+Δt, we, k ). 
 
   As discussed above, each instance of RightArea or LeftArea may be computed using three possible techniques, depending on the region in which t falls. Thus, a maximum of nine combinations of algorithms must be considered, and the circumstances in which each of them holds must be identified. Because t can only move rightward through the regions of the engagement, three of these combinations cannot occur. Primed variables (s′, ps′, pe′, we′) refer to the shrunk engagement. Each combination has at most three subcategories. Two of these occur when the original and shrunk engagement are both on the same side of the s=k phase transition (one subcase for s&lt;k and one for s&gt;k), and the third occurs when the original engagement has s&gt;k and the shrunk engagement has s′&lt;k. In some cases, a particular subcategory may not occur. The subcategories permit resolution of the nonlinear functions (Min, Max, and absolute value) in the basic equation. Further simplification is possible by noting that:
 
 s′=we−Δt−ws−k=s−Δt.  
 
   Operationally, the machine scheduling agent  410  attempting to resolve an ROV knows the area needed, the maximum area available from a candidate engagement, where t falls in that engagement, and what the state of s is for the engagement. The decision process of the machine scheduling agent  410  is driven by these variables, and the following discussion is organized around this decision sequence. 
   Case 1: ws&lt;t&lt;ps (shrink left) 
   This case is illustrated in FIG.  18 . There are two subcases, depending on whether t&lt;ps′or t&gt;ps′. Because t&lt;ps, the formula for RightArea(t, ws, we, k) can be expressed as:
 
 RightArea ( t, ws, we, k )= k −( t−ws ) 2 /(2 s ). 
 
The formula for RightArea(t, ws, we′, k) depends on whether t&lt;ps′ or t&gt;ps′. For the subcase t&lt;ps′; the RightArea is:
 
 RightArea ( t, ws, we′, k )= k− ( t−ws ) 2 /(2 s′ ). 
 
However, to reach the subcase t&gt;ps′, ps′ must have moved to the left, so p′=1, pe′=ws+k, and t is in the plateau, yielding:
 
 RightArea ( t, ws, we k )= pe′−t+s′/ 2 =ws+k−t+s′/ 2. 
 
This equation can be evaluated to determine the maximum area available in a given region. For example, the equation used above:
 
 AreaNeeded=RightArea ( t, ws, we, k )− RightArea ( t, ws, we−Δt, k ), 
 
can be evaluated when ps′=t. If the AreaNeeded is less than that value, the case t&lt;ps′ is present. If AreaNeeded is larger than that value, the equation can be further evaluated when ps′=ws (s′=0). If AreaNeeded is less than this value, the case t&gt;ps′ is present. Otherwise, even if the window is shrunk to its kernel size, the required area cannot be recovered, and other sources must be found if additional area is needed.
 
Subcase 1.1: t&lt;ps′.
 
The required area equation becomes:
 
 AreaNeeded=k −( t−ws ) 2 /(2 s )−[ k− ( t−ws ) 2 /(2 s′ )]=( t−ws ) 2 /(2 s′ )−( t−ws ) 2 /(2 s )=( t−ws ) 2 ( s−s′ )/(2 s*s ′) 
 
Replacing s′ by s−Δt and solving for Δt yields: 
         Δ   ⁢           ⁢   t     =         2   ·     s   2     ·   AreaNeeded           (     t   -   ws     )     2     +     2   ·   s   ·   AreaNeeded         .         
         Example: RightArea(2, 1, 10, 2)−RightArea(2, 1, 9, 2)=0.012 at Δt−1.
 
Subcase 1.2: t&gt;ps′.
 
The area needed equation becomes:
 
 AreaNeeded=k− ( t−ws ) 2 /(2 s )−[ ws+k−t+s′ /2 ]=k− ( t−ws ) 2 /(2 s )−( ws+k−t+ ( s−Δt )/2)=−( t−ws ) 2 /(2 s )−( s−Δt )/2 +t—ws.  
 
Replacing s′ by s−Δt and solving for Δt yields: 
         Δ   ⁢           ⁢   t     =       2   ·   AreaNeeded     +           (     s   -   t   +   ws     )     2     s     .           
   Example: RightArea(2, 1, 10, 2)−RightArea(2, 1, 3.5, 2)=0.68 at Δt=6.5.
 
Case 1′: ws&lt;t&lt;ps (shrink right).
       

   The maximum area that can be recovered is when ws′=t. If the maximum area available is less than the AreaNeeded, other sources must be found if additional area is needed. The equations for LeftArea(t, ws, we, k) and LeftArea(t, ws′, we, k) are given by:
 
 LeftArea ( t, ws, we, k )= p*t−ws )/2=( t−ws ) 2 /(2 *s ); and 
 
 LeftArea ( t, ws′, we, k )  p ′*( t−ws ′)/2=( t−ws−Δt ) 2 /(2*( s−Δt )). 
 
The area needed equation becomes:
 
 AreaNeeded =( t−ws ) 2 /(2 *s )−( t−ws−Δt ) 2 (2*( s−Δt)).  
 
The quadratic equation of Δt, may be simplified to yield:
 
 s*Δt   2 −[2 *s*AreaNeeded +( t−ws )*( ws+ 2 *s−t )]*Δ t+ 2 *s   2   *AreaNeeded= 0, 
 
and solved to yield: 
           Δ   ⁢           ⁢   t     =       a   -         a   2     -     8   ·     s   3     ·   AreaNeeded             2   ·   s         ,       
 
where
 
 a= 2 ·AreaNeeded +( t−ws )·( ws+ 2 ·s−t ). 
 
Case 2: ps&lt;t&lt;pe (shrink left).
 
For this case, it must be distinguished whether s&gt;k or s&lt;k in the original engagement.
 
Subcase 2.1: s&gt;k
 
     FIG. 19  illustrates the three subregions in this subcase. As the engagement shrinks, first t meets pe′ and falls off the end of the plateau. Subsequently s′=k, and, finally, t runs into we′, and the engagement no longer intersects the ROV. If t is in the plateau with s&gt;k, it is to the right of ps=ws+k. When s′=k, then ps′=pe′=ws+k, so pe′ has already passed t. 
   Subcase 2.1: ps&lt;t&lt;pe, s&gt;k, (Example: t=4, ws=1, we=10, k=2). 
   At the end of region 1, =pe′=we′−k, so we′=t+k, and the maximum area available is:
 
RightArea(t, ws, we, k)−RightArea(t, ws, t+k, k). 
 
The end of region 2 occurs when s′=k, or we′ws+2*k, so the area available is:
 
RightArea(t, ws, we, k)−RightArea(t, ws, ws+2*k, k). 
 
Region 3 ends when t=we′, yielding the maximum area:
 
RightArea(t, ws, we, k)−RightArea(t, ws, t, k). 
 
Each of these three regions is considered in turn.
 
Subcase 2.1.1: s′&gt;k, t&lt;pe′.
 
The AreaNeeded Equation becomes:
 
 AreaNeeded =( k/s )*[( we−k− t)+ k/ 2 ]−k /( s−Δt )*[( we−k−Δt−t )+ k/ 2], 
 
which can be solved to yield: 
         Δ   ⁢           ⁢   t     =           s   2     ·   AreaNeeded         s   ·   AreaNeeded     +     k   ·     (       k   /   2     +   ws   -   t     )           .         
         Example: RightArea(5,1,10,2)−RightArea(5,1,9,2)=0.143 at Δt=1.
 
Subcase 2.1.2: s′&gt;k, t&gt;pe′.
 
Now t has fallen off the plateau and entered the right ramp. The AreaNeeded Equation becomes:
 
 AreaNeeded =( k/s )*(( we−k−t )+ k/ 2)−( k /( s−Δt ))*( we−Δt−t ) 2 /(2 *k ). 
 
which can be simplified to yield:
 
 s*Δt   2   −[k   2 +2*( t−we )*( k−s )+2 *s*AreaNeeded]*Δt+s*[k   2 +( t−we )*( t+ 2 *k−we )+2 *s*AreaNeeded]= 0, 
 
and solved to yield: 
           Δ   ⁢           ⁢   t     =       a     2   ·   s       -                 a   2     -     4   ·     s   2     ·     [       k   2     +       (     t   -   we     )     ·                         (     t   +     2   ·   k     -   we     )     +     2   ·   s   ·   AreaNeeded                 2   ·   s           ,       
 
where
 
 a=k   2 +2·( t−we )·( k−s )+2 ·AreaNeeded.  
   Example: RightArea(7, 1, 10, 2)−RightArea(7, 1, 8, 2)=0.471 at Δt=2.
 
Subcase 2.1.3: s′&lt;k, t&gt;pe′.
 
The AreaNeeded Equation becomes:
 
 AreaNeeded =( k/s )*[( we−k−t )+ k/ 2]−( we−t ) 2 /(2*( s−Δt )), 
 
which can be solved to yield: 
         Δ   ⁢           ⁢   t     =       A   +       2   ·     (     k   -   s     )       ⁢     (     t   -   we     )       -               -   4     ·     s   2     ·     (       k   2     +                     (     A   +       (     t   -   we     )     ⁢     (       2   ⁢   k     +   t   -   we     )         )     +                 2   ·     (     k   -   s     )     ·     (     t   -   we     )       +                     2   ·   s   ·   AreaNeeded     )     )     2                 2   ⁢   s           
 
where
 
 A=k   2 +2 ·AreaNeeded·s.  
   Example: RightArea(3.5, 1, 10, 2)−RightArea(3.5, 1, 4, 2)=1.446 at Δt=6.       

   Subcase 2.2: s&lt;k. 
   As  FIG. 18  illustrates, if t starts in the plateau with s&lt;k, it never leaves, since pe is fixed at ws+k. So this subcase has only one region, which ends when s′=0 and we′=ws+k. Thus the maximum area available is:
 
RightArea(t, ws, we, k)−RightArea(t, ws, ws+k, k). 
 
   The AreaNeeded equation takes the form:
 
 AreaNeeded=Δt/ 2. 
 
which yields the immediate solution:
 
Δ t= 2 *AreaNeeded.  
         Example: RightArea(2.5, 1, 4, 2)−RightArea(2.5, 1, 3.5, 2)=0.25 at Δt=0.5.
 
Case 2′: ps&lt;t&lt;pe (shrink right).
 
Similarly, it must be distinguished whether s&gt;k or s&lt;k in the original engagement.
 
Subcase 2′.1: s&gt;k
       

     FIG. 20  illustrates the three subregions in this subcase. As the engagement shrinks, t first meets ps′ and then falls off the end of the plateau. Subsequently s′=k, and, finally, t runs into ws′, and the engagement no longer intersects the ROV. If t is in the plateau with s&gt;k, it is to the left of pe=we−k. When s′=k, then ps′=pe′=we−k, so ps′ has already passed t. 
   Subcase 2′. 1: ps&lt;t&lt;pe, s&gt;k (example: t=4, ws=1, we=10, k=2). 
   At the end of region 1, t=ps′=ws′+k, so ws′=t−k, and the maximum area available is:
 
LeftArea(t, ws, we, k)−LeftArea(t, t−k, we, k). 
 
The end of region 2 occurs when s′=k, or ws′=we−2*k, so the area available is:
 
LeftArea(t, ws, we, k)−LeftArea(t, we−2*k, we, k). 
 
Region 3 ends when t=ws′, yielding the maximum area:
 
LeftArea(t, ws, we, k)−LeftArea(t, t, we, k). 
 
Each of these regions is considered in turn.
 
Subcase 2′.1.1:s&gt;k, t&gt;ps′.
 
The AreaNeeded equation takes the form:
 
 AreaNeeded =( k/s )*[( t−ws−k )+ k /2 ]−k /( s−Δt )*[( t−ws−Δt−k )+ k /2], 
 
which can be solved to yield:
 
Δ t= 2 *AreaNeeded*s   2 /( k   2 +2 *AreaNeeded*s+ 2 *k *( ws+s−t ), 
 
or 
         Δ   ⁢           ⁢   t     =         2   ·     s   2     ·   AreaNeeded         2   ·   s   ·   AreaNeeded     -     k   ·     (     k   -     2   ·   we     +     2   ·   t       )           .         
 
Subcase 2′.1.2: s&gt;k, t&lt;ps′.
 
Now t has fallen off the plateau and entered the left ramp, and AreaNeeded equation takes the form:
 
 AreaNeeded= ( k/s )*(( t−ws−k )+ k/ 2)−( t−ws−Δt ) 2 /(2*( s×Δt )), 
 
which can be simplified to yield:
 
 s*Δt   2 −[2 *s*AreaNeeded+k   2 +2*( t−ws )*( s−k )]*Δ t+s *[( ws+k−t ) 2 +2 *s*AreaNeeded]= 0, 
 
and solved to yield: 
         Δ   ⁢           ⁢   t     =                 k   2     +     2   ·                     (     t   -   ws     )     ·     (     s   -   k     )       +               2   ·   s   ·   AreaNeeded             2   ·   s       -               [       k   2     +     2   ·     (     t   -   ws     )     ·     (     s   -   k     )       +                       2   ·   s   ·   AreaNeeded     ]     2     -               4   ·     s   2     ·     [         (     ws   +   k   -   t     )     2     +                     2   ·   s   ·   AreaNeeded     ]               2   ·   s             
 
Subcase 2′.2: s&lt;k
 
   Symmetric to the ROV to Right case, if t starts in the plateau with s&lt;k, it never leaves, since ps is fixed at we−k. So this subcase has only one region, which ends when s′=0 and ws′=we−k. Thus the maximum area available is:
 
LeftArea(t, ws, we, k)−LeftArea(t, we−k, we, k), 
 
and the AreaNeeded equation takes the form:
 
 AreaNeeded=Δt/ 2, 
 
which yields the immediate solution:
 
Δ t= 2 *AreaNeeded.  
 
Case 3: pe&lt;t&lt;we (shrink left).
 
   This is the simplest case. The maximum area we can reduce is when we′=t. If the maximum area available is less than the AreaNeeded, other sources must be found if additional area is needed. The equations for RightArea(t, ws, we, k) and RightArea (t, ws, we′, k) are given by:
 
 RightArea ( t, ws, we, k )= p *( we−t )/2=(we−t) 2 /(2 *s ), and 
 
 RightArea ( t, ws, we, k )=p′*( we′−t )/2=( we−t−t ) 2 /(2*( s−Δt )) 
 
Thus, the AreaNeeded equation becomes:
 
 AreaNeeded =( we−t ) 2 /(2 *s )−( we−Δt−t ) 2 /(2*( s−Δt )), 
 
which may be simplified to yield;
 
 s*Δt   2 −[2 *s*AreaNeeded+ ( we−t )*( t+ 2 *s−we )]*Δ t+ 2 *s   2   *AreaNeeded =0, 
 
and solved to yield: 
         Δ   ⁢           ⁢   t     =       a   -         a   2     -     8   ·     s   3     ·   AreaNeeded             2   ·   s           
 
where
 
 a= 2· s·AreaNeeded +( we−t )·( t+ 2 ·s−we ). 
         Example: RightArea(8.5, 1, 10, 2)−RightArea(8.5, 1, 9, 2)=0.140 at Δt=1
 
Case 3′:pe&lt;t&lt;we (Shrink Right)
       

   There are two subcases, depending on whether t&lt;pe′or t&gt;pe′.  FIG. 21  illustrates the two subregions in this case. Similar to the cases of ROV to Right, the formula for LeftArea(t, ws, we, k) can be expressed as:
 
 LeftArea ( t, ws, we, k )= k− ( we−t ) 2 /(2 *s ). 
 
The formula for the LeftArea(t, ws′, we, k) depends on whether t&lt;pe′ or t&gt;pe′. For the subcase t&gt;pe′, the LeftArea equation is:
 
 LeftArea ( t, ws′, we, k )=k−( we−t ) 2 /(2 *s ). 
 
For the subcase t&lt;pe′, p′=1, which yields:
 
 LeftArea ( t, ws′, we, k )= k −( s′/ 2+ pe′−t.  
 
As pe′ increases and passes the point t=pe′, the t=pe′ phase transition is reached. The AreaNeeded equation can be evaluated to learn the maximum area available in a given region. For example, the AreaNeeded equation can be evaluated when pe′=t. If AreaNeeded is less than that value, the case t&gt;pe′ is present. If AreaNeeded is larger than that value, the AreaNeeded can be further evaluated when pe′=we(s′=0). If AreaNeeded is less than this value, the case t&lt;pe′ is present. Otherwise, even if the window is shrunk to its kernel size, no more area reduction call occur from this engagement, and another source must be found if additional area is needed.
 
Subcase 3′.1: t&gt;pe′
 
The AreaNeeded equation takes the form:
 
 AreaNeeded=k− ( we−t ) 2 /(2 *s )−[ k −( we−t ) 2 /(2 *s ′)=( we−t ) 2 /(2 *s ′)−( we−t ) 2 /(2 *s ). 
 
Replacing s′ by s−Δt yields:
 
Δ t ( we−t ) 2 −2 s ( s−Δt ) AreaNeeded =0. 
 
Solving for Δt yields: 
         Δ   ⁢           ⁢   t     =         2   ·     s   2     ·   AreaNeeded           (     we   -   t     )     2     +     2   ·   s   ·   AreaNeeded         .         
 
Subcase 3′.2: t&lt;pe′
 
The AreaNeeded equation takes the form:
 
 AreaNeeded=k− ( we−t ) 2 /(2 *s )−[ k−s′/ 2−( pe′−t )]= s′   /2+(   we−s′−t )−( we−t ) 2 /(2 *s ). 
 
Replacing s′ by s−Δt, and simplifying the equation yields: 
           Δ   ⁢           ⁢   t     -         (     s   +   t   -   ws     )     2     s     -     2   ·   AreaNeeded       =   0.       
 
Solving for Δt yields: 
         Δ   ⁢           ⁢   t     =       2   ·   AreaNeeded     +           (     s   +   t   -   ws     )     2     s     .           
 
This completes the discussion of the equations needed to compute the time differential to be used when shifting or shrinking the working window of an engagement to achieve a desired area reduction for an ROV
 
   Returning to  FIG. 4 , the discussion now shifts to how the machine scheduling agent  410  decides which of the engagements are to be executed. When a process tool  115  is idle (or is about to become idle), the machine scheduling agent  410  selects an engagement from its schedule of engagements to execute. Before making this selection, the machine scheduling agent  410  may wish to merge several engagements into one optimized for batching and/or setup. If the process tool  115  is not capable of processing batches, only setup optimization is needed. If the process tool  115  is capable of processing batches, two lots  130  may differ in setup type and/or batch membership. If two lots  130  are in the same batch, they necessarily have the same setup. Thus, there are only three cases for lots  130  on a process tool  115  capable of batching: same batch, different batch but same setup (for example, when two lots  130  must run for different periods of time), and different batch with different setup. 
   The discussion of this operation of the machine scheduling agent  410  begins with batching and setup mergers, and then describes how the next engagement (whether atomic or merged) is selected to run.  FIG. 22  illustrates the effects of joining two lots  130  into a single batch. To be eligible for a batching process, the kernels, k, of the lots  130  to be batched are equal in length to one another and to the kernel of the resulting batch. For two lots, a and b, to be batched together without violating either of their working windows, wws(x), wwe(x), there must be enough time between the latest of their start times and the earliest of their end times to complete the kernel. In terms of the notation of  FIG. 22 , this requirement is:
 
Min( wwe ( a ),  wwe ( b ))−Max( wws ( a ),  wws ( b ))≧ k  
 
   The result of joining two lots  130  under this condition is a batch with the following window limits:
         wws(a+b)=Max(wws(a), wws(b))   wwe(a+b)=Min(wwe(a), wwe(b))       

   In some cases, as discussed below, it may be desirable to violate the window limits of a lot  130  in order to make up a batch, and in these cases the limits would not apply. 
   To grow a batch, the machine scheduling agent  410  selects a batch type to grow, a seed time, t 0 , and a seed engagement whose working window includes t 0 . The machine scheduling agent  410  then adds engagements one by one until the batch is full or there are no more candidate engagements. In an alternative embodiment, a second layer of bidding may be introduced. After lots  130  have successfully bid for access to a resource, they would bid for access to a batch. In the following discussion, the seed technique is discussed. Implementing the seed strategy involves the definition of techniques for selecting the batch type and t 0 , for selecting among alternative candidates for the seed, and for selecting successive engagements to add to the batch. 
   Two exemplary approaches to selecting the batch type and to include one that looks ahead over a fixed time horizon to look for candidate engagements and a second that sets the horizon dynamically Both cases take into account the committed capacity for each possible batch type and a normalized process-operation budget (PO_Budget) for that type, which is the PO_Budget divided by the kernel time. In general, types that have higher cc batchType  are favored to form larger batches and thus increase process tool utilization. In addition, higher normalized process-operation budgets are also favored to accommodate higher priority lots. If all lots  130  of a given batch type had the same normalized process-operation budget, the revenue of the process tool  115  could be maximized by choosing the batch type that maximizes PO_Budget batchType *cc batchType /kernel batchType . The committed capacity is divided by the kernel width because cc is normalized by time, while the PO_Budget is not. Each lot  130  of a given batch type may have a different PO_Budget, due to differences in priority, so the process tool  115  uses an average PO_Budget over the lots  130  of the specified batch type:
 
 BatchPriority   batchType ( t )= cc   batchType ·Σ( PO   —   Budget )/( numLots   batchType ( t )· kernel   batchType ) 
 
where the sum is over lots of the given batch type within whose working windows t falls.
 
   Using the fixed horizon approach, the machine scheduling agent  410  looks ahead over a predefined time horizon and selects the batch type that has the highest BatchPriority within that horizon. The machine scheduling agent  410  then selects to as the point in time within the horizon for which BatchPriority(t) is maximum. The time horizon over which the search is conducted reflects the length of time the process tool  115  can be left idle, since on average to can be expected to fall in the center of this time period. 
   Using the dynamic horizon approach, the machine scheduling agent  410  sets to t 0  the time when the resource is expected to be available, and identifies the batch type as the type with the highest BatchPriority(t 0 ). The machine scheduling agent  410  then looks ahead over the kernel width of this batch type (plus setup time, if a new setup is required), and selects the batch type with the highest BatchPriority within that period, resetting to t 0  the time at which this BatchPriority is maximum. 
   The seed engagement is selected from those engagements whose working window includes t 0 . Several alternative selection criteria are possible. Exemplary selection criteria are provided in order of priority as implemented in the illustrated embodiment. Of course other selection criteria and different priorities may be used, depending on the particular implementation.
         Criterion 1: Select the engagement with the highest normalized budget, since it is the highest priority.   Criterion 2: Select the engagement with the earliest wws, to minimize idle time on the machine.   Criterion 3: Select the engagement with the narrowest working window, since it will be the hardest to accommodate in a later batch.   Criterion 4: Select the engagement with the widest working window, since it will generate the largest set of other engagements that will be eligible to join this batch.
 
The machine scheduling agent  410  may also make the choice based on a weighted function of an engagement&#39;s wws, working window width, and normalized budget.
       

   The machine scheduling agent  410  grows the batch by adding engagements until the batch is full or until there are no more candidate engagements. Engagements are added one by one from those candidates that meet the eligibility requirements outlined above. The criteria for selecting among multiple candidate engagements in the illustrated embodiment are qualitatively similar to those used for selecting the seed. An additional criterion that may be considered is selecting the engagement whose working window overlaps the most with the batch&#39;s current working window. This criterion may have a priority between criteria 1 and 2 listed above. However, in some embodiments, the best criterion for growing the batch may not be the same as the best one for initiating it. For example, the seed may be selected based on working window width, but successive additions may be selected based on earliest wws. A weighted combination of criteria, such as normalized lot budget and overlap, may be considered. 
   In cases where the batch cannot be filled at the time it is created, the machine scheduling agent  410  may elect to delay the processing of the batch so that other engagements may be added. An incomplete batch (lacking N lots) can start at wws (which is designated t 0 ), or wait for up to N lots L i  to become available for inclusion at known times t i , i={1 . . . n}. These additional lots  130  may already be scheduled on one of the process tools  115 , but their earliest start times (ESTs) are too late to satisfy the eligibility requirements outlined above. In determining whether to wait for additional engagements to fill up the batch, the machine scheduling agent  410 :
         defines the costs borne by the process tool  115  if it starts the batch at each possible time t i , i={0 . . . n};   defines the costs borne by the lots (both those in the original batch and L i , i={1 . . . n} as well as lots  130  not in the original batch whose engagements are delayed) if the batch starts at each time t i , i={0 . . . n}; and   starts the batch at the t i  for which the total cost (process tool  115  plus lots) is least.       

   In determining the cost borne by the process tool  115 , the machine scheduling agent  410  may assume that the process tool  115  has a known cost per unit time. This cost can either be considered a constant rate R (assumed hereinafter for the sake of simplicity) or a rate R(t) that varies over time depending on the moment-by-moment utilization of the process tool  115  (i.e., similar to the urgency discount described above). The rate, R, has three potential components. 
   1. The capital cost, R capital , is typically amortized over the expected life of the process tool  115 , and is often charged against the lots that use the process tool  115 . However, this is a sunk cost that does not vary with load, and thus is not necessary to factor into decisions about allocating lots to process tools  115 . Accordingly, it may be excluded from the rate determination. 
   2. The operating cost, R operating , is made up of expenses directly related to the load on the individual process tool  115 , and includes consumables, power, and the cost of cycle-dependent PMs. This factor is included in the rate determination. 
   3. The opportunity cost, R opportunity , reflects the cost of making the process tool  115  unavailable to other lots in order to service a given lot, and depends dynamically on the relationships among various process tools  115  in the fab. In general, different opportunity costs may be appropriate for bottlenecks and non-bottlenecks.
         If a process tool  115  is a bottleneck, it restricts the flow of the entire process flow  100 . Any delay that a lot  130  accumulates waiting for such a process tool  115  is added directly to its overall residency time in the fab, and will delay the delivery of that lot&#39;s dollar value at the end of the line. Thus, R opportunity  for a bottleneck process tool  115  may be set as the final dollar value of the fab&#39;s output per unit time. For example, if the fab produces one lot  130  per hour and each lot has a value of $1 M, then the opportunity cost of idling a bottleneck is $1 M/hour.   If a process tool  115  is not a bottleneck, idle time on it is free, until it attracts so much work that it becomes the bottleneck, at which point its R opportunity  becomes the rate of the bottleneck.       

   In practice, such a step function between bottlenecks and non-bottlenecks is likely to introduce instabilities to the system. Accordingly, R opportunity  is assigned to non-bottleneck process tools  115  based on how close they are to being a bottleneck. A “busyness” estimator, b i , may be computed for each process tool  115 , i, in the range [0, 1], as the ratio of some operating parameter on the process tool  115  in question to the comparable parameter on a bottleneck. If the parameter increases with busyness:
 
 b   i =parameter(machine i )/parameter(bottleneck). 
 
If it decreases with busyness:
 
 b   i =parameter(bottleneck)/parameter(machine i ). 
 
Five exemplary parameters that may be used are:
         1. Percent idle time (bottleneck/machine i );   2. The percent utilization (machine i /bottleneck);   3. 1-percent idle time (like utilization, but accounts for differences in downtime and PM) (machine i /bottleneck);   4. The ratio between average batch size and full batch size for batching machines (machine i /bottleneck); and   5. The input queue length (machine i /bottleneck).
 
The opportunity cost is then:
 
 R   opportunity   =R   opportunityBottleneck   b   i   a , 
 
where a≧1 is a tuning parameter that determines the convexity of the curve relating busyness to rate. For a=1, the rate increases linearly with busyness, while for a&gt;1, the curve is convex, and increasingly so as a increases.
 
Finally, the overall rate is defined as:
 
 R=R   operating   +R   opportunity . 
       

   The cost incurred by a process tool  115  in waiting until start time t w  to start a batch scheduled to start at time t 0  has two components: 
   1. The wait cost of staying idle until t w  waiting for later lots to arrive is:
 
 MW ( t   w )=( t   w   −t   0 )* R.  
         2. The run cost of running with an incomplete batch results from the need to run another batch later to accommodate lots that could have been processed in this batch. This cost depends on A (average batch size required from the machine), C (batch size), and t b (time to run a batch), and is:
 
 MR ( t   w )=]( C ( t   0 )− C ( t   w ))* t   b   /A]*R.  
    Note that C is a function of time. Note also that if C(t w )&gt;C(t 0 ), this cost is negative, reflecting the fact that the process tool  115  is ahead.
 
The total process tool cost is thus:
 
 M ( t   w )= R *[( t   w   −t   0 )−( t   b   /A* ( C ( t   w )− C (t 0 )))]. 
 
It is worthwhile waiting as long as this value is decreasing, a condition that is satisfied while its first derivative with respect to t w  is less than 0. That is:
 
1= C′ ( t   w )* t   b   /A&lt; 0. 
 
C′( t   w )&gt; A/t   b . 
       

   Batch start decisions may cause two kinds of lots to incur waiting costs. If the batch waits, lots already in the batch may be delayed. Once the batch starts, following lots cannot begin until this batch finishes, and hence they may be delayed. In both cases, lateness costs may be computed using the cost of lateness function, COL(t a ), function defined above. This function defines a multiplier to the PO budget if the lot completes at t a . With floating tasks, the end time is not nailed down, but must be no later than LDT. The lateness cost is always relative to a reference time (e.g., t w  relative to t 0 ). Thus, computing the cost of delaying a lot  130  involves:
         computing COL(t w +k)−COL(t 0 +k); and   multiplying this difference by the process-operation budget (PO_Budget) to get a cost that can be combined with MW and MR. For the purpose of making batching decisions, this budget is adjusted by a configurable factor, COLF, as discussed below:
 
Cost of delay=( COL ( t   w   +k )− COL ( t   0   +k ))* PO   —   Budget*COLF  
       

   Another kind of lot  130  may experience a reduction in waiting costs. Lots  130  whose current start time is t i  and are not currently in the batch because they cannot start at time to, but can be added to the batch if the start is delayed to t w , can finish earlier if t 0 &lt;t w &lt;t i . For these lots  130 , the machine scheduling agent  410  calculates the difference in the cost of lateness function, COL, at the current start time t i  of the lot  130  and the earlier time t w ,
 
COL(t w +k)−COL(t i +k), 
 
and then multiply this result (typically negative) by the PO_Budget and the adjustment factor COLF:
 
Cost of delay=( COL ( t   w   −k )− COL ( t   i   +k ))* PO   —   Budget*COLF.  
 
   Note that the lots  130  that may experience a reduction in waiting costs need not have engagements with the same process tool  115  for which the start or delay decision is contemplated. These lots may have engagements on other process tools  115 , but they can be attracted by machine-initiated negotiation. 
   This computation may be conducted by the lot scheduling agent  405 . The machine scheduling agent  410  may provide the lot scheduling agent  405  with two end times, and the lot scheduling agent  405  may reply with the cost to delay from the first end time to the second end time. Now, this mechanism is applied to each class of delayed lot. 
   Lots currently in the batch incur a wait cost LW(t w ) of staying idle until t w  waiting for later lots to arrive. This cost is equal to the sum of the cost of delay values calculated using the function described above over those lots currently in the batch for which LDT&lt;t w +t b . The run cost LR(t w ) of starting the batch at t w  (whether complete or incomplete) results from the fact that lots not in the batch now cannot start until the batch completes. To compute this value, all lots L j  not in the batch are identified as the set of lots  130  L late  such that t w +t b &gt;LDT j −t b . The effective end time for these lots (and their batches) is thus t x =t w +t b +k i , where k i  is the kernel time for the lot i. Let t i  be the batch start time for L i  before being delayed (which may have been later than t 0 +t b ). The cost is thus:
 
 LR ( t   w )=Σ( COL ( t   x )− COL ( t   i +k i ))· PO   —   Budget·COLF  over the lots,  L   late . 
 
   The budget by which the lateness factor is multiplied may be different for batching decisions than for lot bid evaluation. In lot bid evaluation, the market mechanisms are making a trade-off against other decisions involving this same lot. The cost of the trade-offs is borne by the lot&#39;s PO_Budget, which is thus naturally comparable to other factors of the decision. In making decisions about delaying a batch, the trade-offs involve other lots and resources, not just one lot, and the cost is borne by the entire fab. As a result, the “Budget” figure in the previous paragraphs (the batching budget) is adjusted by a configurable factor COLF. 
   The waiting cost values may be used to determine whether to wait and how long to wait before starting the batch. For each t i , i={0 . . . n}, the cost, Cost(t i )=MW(t i )+MR(t i )+LW(t i )+LR(t i ), is determined and the start time, t i , for which this cost is minimum is selected. Note that the best solution may be to start the partial batch at the currently scheduled start time, t 0 , rather than waiting. 
   This estimate of the cost of waiting considers only the effect of the delay on the batch immediately after the batch we are currently forming. On a heavily loaded resource, delaying that latter batch may in turn delay subsequent batches, leading to a ripple effect. This ripple effect varies with the distance into the future over which the process tool  115  is committed to specific lots. The ripple effect increases in severity if the time to execute a batch t b  is short in comparison with the time window over which lots look ahead to schedule, but relatively insignificant if t b  is long in comparison with look-ahead times. 
   The machine scheduling agent  410  may also construct setup chains for batching or non-batching process tools  115 . A setup chain is a group of engagements with the same setup requirements processed sequentially. In the case of batching process tools  115 , batches should be formed first, as outlined in the previous section. Then, these batches may be treated as individual lots for the purposes of setup optimization. One method for constructing setup chains is real-time clustering. When the currently executing engagement is almost finished, the process tool  115  looks for another lot  130  of the same setup type. If a lot  130  of the same setup type is available, the machine scheduling agent  410  selects it for subsequent execution. This simple approach does not permit balancing setup costs against other costs. 
     FIG. 23  illustrates the effects of combining two lots with a common setup. The kernel, k(x), of each independent lot  130  does not contain a setup, s, since only one setup is needed for multiple lots of the same type in sequence, and only the machine scheduling agent  410  can determine which lots can be thus sequenced. As shown in  FIG. 23 , the machine scheduling agent  410  combines sequential lots of the same type into a single chain whose kernel is the sum of one setup and two processing times. Let t setup  be the time required for setup, and k be the processing time. The available time between wws of the earlier lot  130  and wwe of the later lot  130  must be enough to contain this combined kernel:
         Requirement 1: wwe(b)−wws(a)≧t setup +k(a)+k(b).       
   To avoid idle time on the process tool  115 , the setup and processing intervals should follow one another without interruption. In some cases, a gap, MaxGap&lt;t setup , may be permissible. This condition requires that the two working windows overlap:
     Requirement 2: wwe(a)+MaxGap&gt;wws(b).   

   It also restricts the start and end times of the new working window for the combined lots:
     Requirement 3: wws(a+b)≧wws(b)−(t setup +k(a))−MaxGap.   Requirement 4: wwe(a+b)≦wwe(a)+k(b)+MaxGap.
 
For reasons of scheduling flexibility, it is desirable to make the windows as wide as possible. In this case, the new window limits are defined by replacing the inequalities in Requirements 3 and 4 with equalities.
 
Under these restrictions, the new working window has a width of:
 
 wwe ( a+b )− wws ( a+b )≦ wwe ( a )+ k ( b )− wws ( b )+ t   setup   +k ( a )+2 *MaxGap.  
   

   The machine scheduling agent  410  may grow setup chains by selecting a seed lot  130  and repeatedly adding one or more new lots with the same setup type. The machine scheduling agent  410  terminates the process when either of two conditions is satisfied:
         1. There is no eligible engagement whose addition would reduce the net cost; or   2. Further extension of the setup chain would cause a lot  130  with a different setup type to be late.       

   As with batching, an alternative approach would be to let lots  130  bid for access to a setup chain. While more flexible, this approach also increases complexity. 
   To grow a setup chain the machine scheduling agent  410  implements criteria for selecting the seed and subsequent lots. The following discussion outlines several possible heuristics for growing the setup chain. Exemplary heuristics include:
         choosing the lot  130  with the highest normalized budget (PO_Budget/kernel), since that is the one least likely to be bumped;   choosing the lot  130  with the earliest wws;   choosing the lot  130  with the earliest wwe, which is the most likely to be made late (however, addition of this lot shrinks the window for the resulting chain to end at wwe, making it more difficult to find another lot  130  to include in the chain. This heuristic is useful if there are many overlapping engagements from which to choose at the next step); or   choosing the lot  130  with the largest kernel, thus extending the window of opportunity for later engagements to join (This heuristic is useful if subsequent engagements are sparse).
 
If the relatively simple heuristic of choosing the lot  130  with the earliest wws is not chosen, the machine scheduling agent  410  may construct a setup chain where wws(new)&lt;wws(old), which might result in the need to shuffle kernels after the initial setup. However, the earliest lot  130  might have a window that is too narrow to reach later lots and enable them to join the chain.
       

   Choosing one general rule for making setup chain decisions typically will not address the varying situations expected. The most effective rule depends on factors, such as the empirical state of the schedule of engagements for the process tool  115  or the current priorities of the fab and/or the process tool  115 , and may be determined experimentally. As with batching optimization, the heuristic that proves most useful for selecting the initial seed may not be the same heuristic chosen for selecting subsequent additions to the setup chain. 
   In summary, the technique for growing a setup chain includes the following steps, which are illustrated using a pseudo-code representation:
         Select the seed and add the setup time to its kernel (the resulting engagement is called a “chain”)   Loop 1: Repeat   Select new candidate engagement to merge. Two cases are possible:   Case 1: Add at end of existing chain if:   [wws(new)≦wws(chain)+k(chain)+maxGap] &amp; [wwe(new)≧wwe(chain)+k(new)].   Case 2: Add at beginning of existing chain if:   [t 0 ≦wws(new)≦wws(chain)−k(new)] &amp; [wwe(new)≧wws(chain)−maxGap]   If no such engagement exists for Case 1 or Case 2, then exit Loop 1.   If Cost(chain+new)&lt;Cost(chain)+Cost(new)+Cost(setup) (heuristic: if chains are formed at the last minute before execution, replace this condition with True, since there is no longer any need for cost competition among engagements for the same period of process tool  115  availability)
           If adding this lot  130  would cause an engagement of different type to be late, exit Loop 1.   Add this lot  130  to the chain of the previous lots;   Adjust wws, wwe, and k to account for addition. The adjustment depends on the case to which the new engagement belongs.   Case 1:
               wws(next)=wws(new)−k(chain);   wwe(next)=wwe(chain)+k(new);   k(next)=k(chain)+k   
               Case 2:
               wws(next)=wws(chain)−k(new);   wwe(next)=wwe(chain);   k(next)=k(chain)+k(new)   
               Rename “next” to “chain”   
           If any engagements of Case 2 have been added, sort kernels in the chain in order of their original wws times.       

   The technique outlined above terminates if adding the next lot  130  would cause a lot  130  of a different setup type to be late. However, if a setup is particularly costly, it may cost less to make another lot  130  late than to interrupt the setup chain. 
   Consider the case where a chain of type a is being run The cost of the setup, Cost(setup a ), is known. Another lot  130  of type a could be added, but doing so would make a lot  130  of a different type late. The cost of such a trade-off can be computed using mechanisms similar to those described above for batching decisions. The cost of lateness function, COL(t a ) defined above may be used. This function defines a multiplier to the PO_budget if the lot ends at t a . For a floating task, the end time is not nailed down, but must be no later than LDT. Lateness cost is always relative to a reference time (e.g., t w  relative to t 0 ). Thus, to compute the cost of delaying a lot from end time to t 0  end time t w :
         compute COL(t w )−COL(t 0 ); and   multiply this difference by the process-operation budget, PO_Budget, to get a cost that can be compared with Cost(setup a ). (As with batching decisions, this budget may be adjusted by a configurable factor.)
 
Again, this computation may be performed by the lot scheduling agent  405 . The machine scheduling agent  410  may provide the lot scheduling agent  405  with two end times, and the lot scheduling agent  405  may reply with the cost to delay from the first time to the second time.
       

   The discussion now turns to the matter of selecting the next engagement to run. At some point, the machine scheduling agent  410  has a collection of engagements available for execution. These may be atomic engagements from single lots or merged engagements (either batches or setup chains). The machine scheduling agent  410  selects the next engagement to run from the set of scheduled engagements with working window start times prior to or at the time the process tool  115  is expected to become available. If no engagements start this early, the eligible set consists of the engagement(s) with the earliest wws. Exemplary heuristics for selecting the next engagement to execute are provided below. 
   1. For each candidate engagement, determine the cost of moving it to the position where it would execute next. When an engagement executes, the working window is narrowed to the kernel and any engagements whose working windows overlap with the resulting kernel-size working window must also be moved. Such a cost calculation may include the cost of moving other engagements out of the execution window. 
   2 Choose the engagement with the maximum pWind. Such an engagement has the least flexibility. 
   3. Choose the engagement with the highest normalized budget, i.e., budget per unit of kernel time, since budget is correlated with priority. 
   Of course, different heuristics or combinations thereof may also be used in alternative embodiments. 
   The above discussions have focused on one particular type of consumer agent  305 , the lot scheduling agent  405 , and the lot processing engagements it schedules. However, as previously mentioned, other types of consumer agents  305  may be appropriate if there are additional activities that need to be scheduled on the process tool  115 . Another exemplary consumer agent  305  is a PM/Qual scheduling agent  490  (shown in  FIG. 4 ) that is responsible for scheduling the preventive maintenance procedures that need to be performed periodically on the process tool  115 . The PM/Qual scheduling agent  490  schedules PMs and Quals (qualification procedures). A Qual is a test to ensure that the process tool  115  is performing properly. A PM is a procedure to overhaul and/or replace parts of the process tool  115  so that it will continue to perform properly. 
   These preventive maintenance procedures, PMs and Quals, must be performed periodically, with a frequency that may be based on elapsed time, wafers/lots/batches processed, processing time, event occurrences, degradation of a measurable parameter or other criteria. Although the PMs and Quals generally need to occur at the specified frequency, it is typical to allow some flexibility in the scheduling. One method of specifying this flexibility is to define a Window Start Offset and a Window End Offset. For example, a 30-day PM that takes 12 hours to perform is scheduled at a frequency of 30 days, and has the flexibility to be performed up to 2 days early and as much as 3 days late. For this 30-day PM the frequency is 30 days, the Window Start Offset is 2 days and the Window End Offset is 3 days. The 30-day PM can therefore be scheduled during any contiguous 12-hour period beginning as early as 28 days after the last 30-day PM or ending as late as 33 days after the last 30-day PM on the same machine. 
   Therefore when the PM/Qual scheduling agent  490  requests a bid from the machine scheduling agent  410  to schedule the 30-day PM, it specifies a commitment window bounded by an earliest start time (EST) and a latest delivery time (LDT) that are calculated using the frequency, window start offset, window end offset and date of the last occurrence of the 30-day PM as shown below. Note that the EST and LDT may be adjusted to reflect the current time and the kernel, k (duration of the PM/Qual).
         EST=Max(last occurrence+frequency−window start offset, current time)   LDT=Max(last occurrence+frequency+window end offset, current time+k)       

   Each PM/Qual also has a budget for purchasing lime from the machine to perform the preventive maintenance procedure. The PM/Qual scheduling agent  490  has a PM budget calculator that can be called by the scheduling agents to calculate the budget for a specific PM/Qual at a particular completion time. The PM budget calculator employs a function that considers the time required to perform the PM (duration) as well as the percentage of the commitment window that has transpired by the completion time (window percentage). To achieve effective scheduling, the PM budget provides more funds for a PM/Qual with a longer duration and the budget for a specific PM/Qual also increases as the window percentage increases. In other embodiments, the PM budget may also consider other factors. 
   When sending a bid request to the machine scheduling agent  410 , the PM/Qual scheduling agent  490  provides the following information:
         EST   LDT   Identity of the PM/Qual requesting the bid   PM Budget Calculator       

   The bid returned to the PM/Qual scheduling agent  490  by the machine scheduling agent  410  is similar to the bid returned to the lot scheduling agent  405 , with some simplification: there is no potential for batching or setup optimization, since a PM/Qual cannot be “batched” with any other activity scheduled on the machine and a PM/Qual has no “setup”. So the bid from the machine scheduling agent  410  contains:
         BCF=Basic Cost Function for the time window [EST,LDT]   ccSameSetup avg =0   RTU ccDiff =Rate per unit-time function based on committed capacity, with   ccDiffSetup fixed at cc avg , the average committed capacity of all engagements within the time window [EST,LDT] (in this case all engagements are not the same type)   FDF=Flexibility discount function       

   The PM/Qual scheduling agent  490  uses the same algorithm as the lot scheduling agent  405  to generate a collection of candidate bids by sampling the BCF for commitment windows with varying sizes, start times and end times. The PM/Qual scheduling agent  490  also calculates the cost of each candidate bid, again using the same algorithm as the lot scheduling agent  405 , with the batching factors eliminated (ccSameBatch avg =0). 
   The PM/Qual scheduling agent  490  then evaluates the candidate bids based on its objective function, which considers the cost of the bid as well as the end (completion) time. The PM/Qual scheduling agent  490  chooses the bid that minimizes the value of the objective function. In one embodiment, the PM/Qual scheduling agent  490  selects the earliest affordable bid, i.e. the earliest bid that can be afforded with its applicable budget for the bid end time. 
   The PM/Qual scheduling agent  490  then asks the machine scheduling agent  410  to confirm the selected bid. Using the same algorithm as it does when confirming a bid for a lot scheduling agent  405 , the machine scheduling agent  410  recalculates the cost of the selected bid based on the current schedule of engagements for the process tool  115 . If the cost does not exceed a configurable percentage of the original cost, the bid is confirmed, otherwise it is denied. If the PM/Qual scheduling agent  490  receives a denial, it uses the same algorithm as the lot scheduling agent  405  to determine whether it should attempt to confirm the next best bid or begin over again by requesting new bids with a wider commitment window. A rebid may be initiated due to the remaining bids falling below a configurable RebidThreshold or if the value of the objective function degrades more than a configurable percentage of the value corresponding to the best bid. If a rebid is initiated, the commitment window is widened by increasing the LDT. In one embodiment, the LDT is increased by a configurable percentage of the kernel, k. 
   As a result of confirming bids for PMs and/or Quals, the schedule of engagements for the process tool  115  may contain engagements for PMs and Quals as well as lots. When calculating committed capacity and costs for lot bids, the engagements corresponding to PMs or Quals are consider a different “type” than any lot bid, and therefore are included in the committed capacity for lots with different setups and/or different batch criteria. When calculating committed capacity and costs for PM/Qual bids, each PM or Qual engagement is considered a different “type” than any other PM, Qual or lot engagement, and therefore every scheduled engagement for the process tool  11 S is included in the committed capacity for different setups and/or different batch criteria. 
   One other type of engagement that can occur in the scheduled engagements for a process tool  115  is a downtime engagement. A downtime engagement is not scheduled in advance; it is scheduled by the machine scheduling agent  410  when the process tool  115  experiences the onset of unscheduled downtime. Unscheduled downtime consumes 100% of the capacity of the process tool  115  and prevents any other activities from occurring until the process tool is repaired. When the machine scheduling agent  410  detects the onset of unscheduled downtime, it schedules a downtime engagement starting at the current time. The kernel time of the downtime engagement is set to the mean-time-to-repair (MTTR) and the commitment window [EST, LDT] is set to the kernel. If there are any other engagements scheduled in this region, a region of violation (ROV) will be created and the machine scheduling agent  410  will use the techniques described earlier to shift, shrink or cancel other engagements in order to resolve the ROV. 
   Note that some portions of the detailed descriptions herein are presented in terms of a software implemented process involving symbolic representations of operations on data bits within a memory in a computing system or a computing device. These descriptions and representations are the means used by those in the art to most effectively convey the substance of their work to others skilled in the art. The process and operation require physical manipulations of physical quantities. Usually, though not necessarily, these quantities take the form of electrical, magnetic, or optical signals capable of being stored, transferred, combined, compared, and otherwise manipulated. It has proven convenient at times, principally for reasons of common usage, to refer to these signals as bits, values, elements, symbols, characters, terms, numbers, or the like. 
   It should be borne in mind, however, that all of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels applied to these quantifies. Unless specifically stated or otherwise as may be apparent, throughout the present disclosure, these descriptions refer to the action and processes of an electronic device, that manipulates and transforms data represented as physical (e.g., electronic, magnetic, or optical) quantities within some electronic device&#39;s storage into other data similarly represented as physical quantities within the storage, or in transmission or display devices. Exemplary of the terms denoting such a description are, without limitation, the terms “processing,” “computing,” “calculating,” “determining,” “displaying,” and the like. 
   Note also that the software implemented aspects of the invention are typically encoded on some form of program storage medium or implemented over some type of transmission medium. Examples of program storage media, without limitation, are magnetic (e.g., a floppy disk or a hard drive), optical (e.g., a compact disk read only memory, or CD ROM), electrostatic/capacitive, tunneling electro microscope, or some other form, and may be read only, read/write, random access, serial access, etc. Similarly, the transmission medium may be twisted wire pairs, coaxial cable, optical fiber, or some other suitable transmission medium known to the art. The invention is not limited by these aspects of any given implementation. 
   The particular embodiments disclosed above are illustrative only, as the invention may be modified and practiced in different but equivalent manners apparent to those skilled in the art having the benefit of the teachings herein. Furthermore, no limitations are intended to the details of construction or design herein shown, other than as described in the claims below. It is therefore evident that the particular embodiments disclosed above may be altered or modified and all such variations are considered within the scope and spirit of the invention. Accordingly, the protection sought herein is as set forth in the claims below.