Statistics, modeling and simulation

In chapter 4 we talked about the significance and the role of statistics in the DMAIC model of the six sigma methodology. In this chapter we are focusing on the statistical role in the DCOV model. The reader will notice that statistics, modeling, and simulation play a much more important role in the design for six sigma as applied through the DCOV model.

Especially in DFSS we must be cognizant of parametric and non-parametric statistics and their application. The term parameter is generally employed to connote a characteristic of the population. A parameter is often an unspecified constant appearing in a family of probability distributions, but the word can also be interpreted in a broader sense to include almost all descriptions of population characteristics within a family. In other words, they are distribution dependent.

The term non-parameter, on the other hand, is a distribution-free inference, and the methods used are based on functions of the sample observations whose corresponding random variable has a distribution that does not depend on the specific distribution function of the population from which the sample was drawn. In other words, assumptions regarding the underlying population are not necessary for testing or estimation.

DFSS by definition is a planning activity, to plan in advance, as much as possible, "flawless" designs that meet or exceed customer expectations. Toward that end robustness is utilized in the form of Y = f(x, n), and modeling is used to predict outcomes. Typical modeling techniques are parameter design, regression, MANOVA (multiple analysis of variance), structural modeling and tolerance design.

Furthermore, we must also consider how a system in the design phase will perform once it is built. Questions include: Will it be stable? Is the control system adequate? How can the performance of an existing system be improved? These are the kinds of tough questions faced every day by engineers working with complex dynamic systems in a broad range of industries—automotive, off-highway equipment, transportation, aerospace, defense, health care, education and so on. Traditionally, solutions to these questions have been found by building costly prototypes and pilots, as well as performing extensive laboratory tests.

Today, especially organizations with a DFSS commitment may reduce their time to market and their development costs using simulation. Simulation is a tool that allows the experimenter to see "what happens if...". For simulation to be effective, the experimenter must have technical expertise and must know the simulated process. Conducting any simulation, however, is just a means to an end, not an end in itself. Common simulations are based on Monte Carlo and numerical approximation methods such as finite element analysis, root sum of squares (RSS) method, successive linear approximation method (SLAM), Taguchi's tolerance design and others.

An effective simulation starts with a model—that is, a set of equations that accurately characterizes system dynamics—and it is here that the big problems begin. Deriving system equations is difficult and confounds even the best engineers. (Sometimes it is so difficult that many companies give up on simulation altogether.) When there is a real difficulty in deriving equations, we can still use simulations with either surrogate data or similar designs and historical experiences. To be sure, the results will be approximations and must be adjusted accordingly as more information is gained through our experimentation.

Reliability

Reliability is valued by the organization and is a primary consideration in all decision-making. Reliability techniques and disciplines are integrated into system- and component-planning, design, development, manufacturing, supply, delivery and service processes. The reliability process is tailored to fit individual business unit requirements and is based on common concepts that are focused on producing reliable products or services and systems, not just components.

In pursuing DFSS, an organization should have a broad statement that frames the overall task and is deployed within the organization. The reliability process must include robustness concepts and methods that are integrated into the organizational design culture and are in tandem with the customer's needs, wants and expectations.

Reliability can be defined simply as the probability that a system or product will perform in a satisfactory manner, for a given period of time, when used under specified operating conditions. This definition stresses the elements of probability, satisfactory performance, time and specified operating conditions. These four elements are extremely important, since each plays a significant role in determining system or product reliability.

Probability. Probability, the first element in the reliability definition, is usually stated as a quantitative expression representing a fraction or a percent signifying the number of times that an event occurs (successes), divided by the total number of trials.

Satisfactory performance. Satisfactory performance, the second element in the reliability definition, indicates that specific criteria must be established that describe what is considered to be satisfactory system operation. A combination of qualitative and quantitative factors defining the functions that the system or product is to accomplish, usually presented in the context of a system specification, is required.

Time. The third element, time, is one of the most important, since it represents a measure against which the degree of system performance can be related. One must know the time parameter in order to assess the probability of completing a mission, or a given function, as scheduled. Of particular interest is being able to predict the probability of an item surviving (without failure) for a designated period of time (sometimes designated as "R"). Also, reliability is frequently defined in terms of mean time between failure (MTBF), mean time to failure (MTTF), or mean time between maintenance (MTBM); thus, the aspect of time is critical in reliability measurement.

Specified operating conditions. The specified operating conditions, under which we expect a system or product to function, constitute the fourth significant element of the basic reliability definition. These conditions include environmental factors such as geographical location where the system is expected to operate, the operational profile, the transportation profile, temperature cycles, humidity, vibration, shock and so on. Such factors must not only address the conditions for the period when the system or product is operating, but the conditions for the periods when the system (or a portion thereof) is in a storage mode or being transported from one location to the next. Experience has indicated that the transportation, handling and storage modes are sometimes more critical from a reliability standpoint than the conditions experienced during actual system operational use.

These four elements are critical in determining the reliability of a system or product. System reliability (or unreliability) is a key factor in the frequency of maintenance, and the maintenance frequency obviously has a significant impact on logistical support requirements. Therefore, reliability predictions and analyses are required as an input to the logistic support analysis.

Reliability is an inherent characteristic of design. As such, it is essential that reliability be adequately considered at program inception, as part of the DFSS process and be addressed throughout the system life cycle.

Robustness and reliability

The traditional six sigma methodology focuses on the DMAIC model which, of course, is based on the notion of Y = f(x). The DFSS methodology is based on the DCOV model and is focused on the notion of Y = f(x, n). Mathematically these two distinct approaches can be shown as:

Whereas the focus of the DMAIC model is to reduce (variability), the focus of the DCOV is to reduce the (sensitivity).

This is very important and that is why we use the partial derivatives of the xs to define the Ys. Of course, if the transformation function is a linear one, then the only thing we can do is to control variability. Needless to say, in most cases we deal with polynomials, and that is why DOE and especially parameter design are very important in any DFSS endeavor. The fact that we introduce the noise in the f(x, n) makes the equation quite powerful, because we are focusing on satisfying the customer regardless of the present noise. In fact, that is what robustness is.

In a more descriptive manner, this robustness may be explained as a 15-step process (shown in Figure 6.3) that reduces complexity, and therefore, variation. The steps are grouped into six sections. The reader will notice that robust, reliable products may be generated from a seed—a vision of unprecedented customer satisfaction, which is the iteration of the 15 simple steps.

Figure 6.3: The flow of a typical robustness approach to reduce complexity

Program input

1. Work as a program team with a shared vision.

   Purpose: establish and maintain a program development team for both product and process that has a shared vision.

   Idea and vision: high-performance teams emerge and grow through the systematic application of learning organization and team disciplines to foster synergy, shared direction, interrelationships and a balance between intrinsic and extrinsic motivational factors. A rapid, focused, simultaneous start-up enables teams to develop a high-quality product on schedule.

2. Create a program information center.

   Purpose: establish and maintain a program information center to understand program, social and institutional knowledge.

   Idea and vision: create an information environment with networks to foster program and/or product knowledge via prior lessons learned, best practices, inter- and intradisciplinary communication and collaboration.

Design architecture

3. Establish and prioritize customer wants, needs and delights.

   Purpose: Identify customers and create opportunities for team members to establish and/or prioritize customer wants, needs, delights, usage profiles and demographics.

   Idea and vision: Foster intense customer engagement to identify base expectations, as well as distinctive opportunities that differentiate and characterize a winning product. Conditions are set that allow the team to get a deep understanding of what is a desirable product from a customer viewpoint. The result is products customers will want to buy.

4. Derive customer-driven specifications.

   Purpose: Translate customer, corporate and regulatory requirements into product/process specifications and engineering and/or test plans.

   Idea and vision: Establish the foundation (maximum potential) for customer satisfaction by systematically translating the customer definition of a "good" product into engineering language and competitive targets. Customer-driven specifications describe a final product that satisfies the real world customer.

5. Define system architecture and function.

   Purpose: Define system architecture, inputs/outputs and ideal function for each of the system elements and identify interfaces.

   Idea and vision: Lay the foundation for analytical optimization of function, cost, quality and performance by gaining understanding of how the system and system elements function ideally, and by gaining understanding of the interfaces and interactions between functional system elements. The engineer has the opportunity to create and innovate high-level architecture to make a significant competitive difference.

Product/process design

6. Select product/process concept.

   Purpose: Create and/or establish alternative product design and manufacturing process concepts and derive enhanced alternatives for development.

   Idea and vision: Derive a concept to meet or exceed customer expectations through systematic exploration of many alternatives. Creative thinking is crucial here. At this point in the process, with only 10 percent of the total product development time committed, 80 percent of the future success is determined.

7. Conduct product and process design.

   Purpose: Design and model product and process concurrently, using low-cost tolerances and inexpensive materials.

   Idea and vision: Achieve superior performance through simultaneous integration of engineering, manufacturing and delivery functions.

8. Prevent failure modes and decrease variability.

   Purpose: Improve product and process through reduction of failure modes and variability.

   Idea and vision: Improve product and process by asking: "what can go wrong?" and "where can variation come from?" Revise design and process to prevent occurrence and reduce variation.

9. Optimize function in the presence of noise.

   Purpose: Optimize product and manufacturing/assembly process functions by testing them in the presence of anticipated sources of variation and/or noise.

   Idea and vision: Improve performance against customer targets, during the development process, by adjusting controllable parameters to minimize deviations from the intended or ideal function.

10. Perform tolerance design.

    Purpose: Selectively tighten tolerances and upgrade materials to achieve desired performance (with cost/benefit trade-offs). Identify key characteristics for manufacturing control and variability reduction.

    Idea and vision: Achieve functional targets at lowest cost by selectively tightening tolerances and upgrading materials only where necessary. Demonstrated customer sensitive characteristics are chosen for ongoing variation reduction. By applying preventive and robust methods early in the design process (steps 7–10), you simultaneously greatly enhance ideal function performance, create a superior product and increase investment and resource efficiency.

11. Finalize process and control plans.

    Purpose: Finalize process and establish tooling, gages and control plans.

    Idea and vision: The manufacturing and assembly processes, tooling, gages and control plans are appropriately designed to control and reduce variation in characteristics that influence customer satisfaction.

12. Verify the design.

    Purpose: Integrate and verify design and manufacturing process functions with production-like hardware and software.

    Idea and vision: Improve quality and reduce time to market by enabling a single prototype build/test/fix cycle. A high-quality, cost-efficient manufacturing process or product is ensured.

Design/manufacturing confirmation

13. Confirm manufacturing capability.

    Purpose: Confirm manufacturing and assembly process capability to achieve design intent.

    Idea and vision: Enable rapid, smooth confirmation of production and assembly operations with minimal refinements. Unanticipated trial concerns are identified and corrected to ensure a low-risk launch.

Launch and mass production confirmation

14. Launch product and ramp-up production.

    Purpose: Launch the product, ramp-up and confirm that mass production delivers function, cost, quality and performance objectives.

    Idea and vision: Implementation of robust processes and up-front launch planning, with just-in-time training, will promote a smooth launch and rapid ramp-up to production speed. Final confirmation of deliverables enables team learning with improved understanding of cause-effect relationships. All the time and effort spent at the beginning of the process now starts to pay off.

For all activities

15. Update corporate memory.

    Purpose: Update the corporate knowledge database with technical, institutional and social lessons learned (both TGR and TGW should be evaluated and logged).

    Idea and vision: Retain what has been learned. Create and maintain capability for locating, collecting and synthesizing data and information into profound knowledge. Communicate, in a timely and user-friendly manner, all technical, institutional and social lessons learned. As we increase our knowledge, we increase our power to become the best.

Maintainability

Maintainability, like reliability, is an inherent characteristic of system, or product, design. It pertains to the ease, accuracy, safety and economy in the performance of maintenance actions. A system should be designed so that it can be maintained without large investments of time, cost or other resources (e.g., personnel, materials, facilities and test equipment) and without adversely affecting the mission of that system. Maintainability is the ability of an item to be maintained, whereas maintenance constitutes a series of actions to be taken to restore or retain an item in an effective operational state. Maintainability is a design parameter. Maintenance is a result of design.

Maintainability can also be defined as a characteristic in design that can be expressed in terms of maintenance frequency factors, maintenance times (i.e., elapsed times and labor hours) and maintenance cost. These terms may be presented as different figures of merit. Therefore, maintainability may be defined on the basis of a combination of factors involving all aspects of the system. The measures of maintainability often include a combination of the following:

• MTBM. Mean time between maintenance includes both preventive (scheduled) and corrective (unscheduled) maintenance requirements. It includes consideration of reliability MTBF and MTBR (see next item). MTBM may also be considered as a reliability parameter.

• MTBR. Mean time between replacement of an item due to a maintenance action (usually generates a spare-part requirement). Items and their symbols associated with MTBR are:

  1. —mean active maintenance time (a function of and ).

  2. —mean corrective maintenance time. Equivalent to mean time to repair (MTTR).

  3. —mean preventive maintenance time.

  4. —median active corrective maintenance time. Equivalent to equipment repair time (ERT).

  5. —median active preventive maintenance time.

  6. MTTR_g—geometric mean time to repair.

  7. M_ax—maximum active corrective maintenance time (usually specified at the 90% and 95% confidence levels).

  8. MDT—maintenance downtime (total time during which a system/equipment is not in condition to perform its intended (function). MDT includes active maintenance time (), logistics delay time (LDT), and administrative delay time (ADT).

  9. MMH/OH—maintenance man-hours; per equipment operating hour.

  10. Cost/OH—maintenance cost per equipment operating hour.

  11. Cost/MA—maintenance cost per maintenance action.

  12. Turnaround time (TAT)—that element of maintenance time needed to service, repair, and/or check out an item for recommitment. This constitutes the time that it takes an item to go through the complete cycle from operational installation through a maintenance shop and into the spares inventory ready for use.

  13. Self-test thoroughness—the scope, depth, and accuracy of testing.

  14. Fault isolation accuracy—accuracy of equipment diagnostic routines in percent.

Maintainability, as an inherent characteristic of design, must be properly considered in the early phases of system development, and maintainability activities are applicable throughout the life cycle.

Supportability

Supportability relates to the degree to which the system can be Supported, both in terms of the inherent characteristics of prime equipment design and the effectiveness of the overall support capability (i.e., elements of logistic support). This term is commonly used in a rather general sense, and its use often implies some degree of overlap with reliability and maintainability.

Serviceability

Serviceability relates to how fast an incapacitated system can be fixed or brought back to service. Serviceability is closely related to human factors—sometimes referred to as ergonomics. Human factors pertain to the human element of the system and the interfaces between the human being, the machine and associated software. The objective is to ensure complete compatibility between the system's physical and functional design features and the human element in the operation, maintenance and support of the system. Considerations in design must be given to anthropometric factors (e.g., the physical dimensions of the human being), human sensory factors (e.g., vision and hearing capabilities), physiological factors (e.g., impacts from environmental forces), psychological factors (e.g., human needs, expectations, attitude, motivation) and their interrelationships. Just as with reliability and maintainability, human factors must be considered early in system development through the accomplishment of functional analysis, operator and maintenance task analysis, error analysis, safety analysis and related design support activities. Operator and maintenance personnel requirements (i.e., personnel quantities and skill levels) and training program needs evolve from the task analysis effort. Maintenance personnel requirements are also identified in the logistic support analysis (LSA).

Maintenance

Maintenance includes all actions necessary for retaining a system or product in, or restoring it to, a serviceable condition. Maintenance may be categorized as corrective maintenance or preventive maintenance:

• Corrective maintenance. This includes all unscheduled maintenance actions performed as a result of system or product failure to restore the system to a specified condition. The corrective maintenance cycle includes failure localization and isolation, disassembly, item removal and replacement or repair, reassembly, and checkout and condition verification. Also, unscheduled maintenance may occur as a result of a suspected failure, even if further investigation indicates that no actual failure occurred.

• Preventive maintenance. This includes all scheduled maintenance actions performed to retain a system or product in a specified condition. Scheduled maintenance includes the accomplishment of periodic inspections, condition monitoring, critical item replacements and calibration. In addition, servicing requirements (e.g., lubrication and fueling) may be included under the general category of scheduled maintenance.

Maintenance level. Corrective and preventive maintenance may be accomplished on the system itself (or an element thereof) at the site where the system is used by the consumer, in an intermediate shop near the consumer, and/or at a depot or manufacturer's plant facility. Maintenance level pertains to the division of functions and tasks for each area where maintenance is performed. Task complexity, personnel-skill-level requirements, special facility needs, and so on, dictate to a great extent the specific functions to be accomplished at each level.

Maintenance concept. The maintenance concept constitutes a series of statements and/or illustrations defining criteria covering maintenance levels. The maintenance concept is defined at program inception and is a prerequisite to system or product design and development. The maintenance concept is also a required input to logistic support analysis (LSA).

Maintenance plan. The maintenance plan (as compared to the maintenance concept) is a detailed plan specifying the methods and procedures to be followed for system support throughout the life cycle during the consumer use period. The plan includes the identification and use of the required elements of logistics necessary for the sustaining support of the system. The maintenance plan is developed from logistic support analysis (LSA) data and is usually prepared during the detail design phase.

System effectiveness

System effectiveness is one or more figures of merit representing the extent to which the system is able to perform the intended function. The figures of merit used may vary considerably depending on the type of system and its mission requirements, and should consider the following:

• System performance parameters. These might include the capacity of a power plant, range or weight of an airplane, destructive capability of a weapon, quantity of letters processed through a postal system, amount of cargo delivered by a transportation system and the accuracy of a radar capability.

• Availability. This is the measure of the degree a system is in the operable and committable state at the start of a mission when the mission is called for at an unknown random point in time. This is often called operational readiness. Availability is a function of operating time (reliability) and downtime (maintainability/supportability).

• Dependability. This is the measure of the system operating condition at one or more points during the mission, given the system condition at the start of the mission (i.e., availability). Dependability is a function of operating time (reliability) and downtime (maintainability/supportability).

A combination of the foregoing considerations (measures) represents the system effectiveness aspect of total cost-effectiveness. By inspection, one can see that logistics impacts the various elements of system effectiveness to a significant degree, particularly in the areas of availability and dependability. System operation is highly dependent on support equipment (handling equipment), operating personnel, data and facilities. Maintenance and system downtime are based on the availability of test and support equipment, spare or repair parts, maintenance personnel, data and facilities. The effect of the type and quantity of logistic support is measured through the parameters of system effectiveness.