Mathematics for Logisticians
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©1999-2003. ReliaSoft Corp
5
Basic System Reliabilitym Reliability Terminology
• The System Reliability Function
– After defining the properties of each block in a system, the blocks can then be connected in a reliability-wise manner to create a reliability block diagram for the system. The RBD provides a visual representation of the way the blocks are reliability-wise arranged. This means that a diagram will be created that represents the functioning state (i.e. success or failure) of the system in terms of the functioning states of its components. In other words, this diagram demonstrates the effect of the success or failure of a component on the success or failure of the system. For example, if all components in a system must succeed in order for the system to succeed, the components will be arranged reliability-wise in series. If one of two components must succeed in order for the system to succeed, those two components will be arranged reliability-wise in parallel. – The reliability-wise arrangement of components is directly related to the derived mathematical description of the system. The mathematical description of the system is the key to the determination of the reliability of the system. In fact, the system's reliability function is that mathematical description (obtained using probabilistic methods) and it defines the system reliability in terms of the component reliabilities. The result is an analytical expression that describes the reliability of the system as a function of time based on the reliability functions of its components.
Mathematics for Logisticians
Introduction and Overview
November 9, 2005
Kenneth East, C.P.L. Frank Vellella, C.P.L.
1
Mathematics in Logistics
• • • • • • • • • • • • • • • • • • Basics of Probability Measures of Central Tendency and Variability Data Description Probability Distributions Series and Parallel Component Reliability Failure Rates Probability of Success with Spares Availability Maintainability Factors Availability Inventory Management-Economic Order Quantity Forecasting-Moving Averages and Exponential Smoothing Regression and Correlation Analysis Queuing Theory Inflation, Discounting and Breakeven Analysis Decision Trees Linear Programming PERT/CPM Learning Curves
2
Basic System Reliability Terminology
The main objective of system reliability is the construction of a model (life distribution) that represents the times-to-failure of the entire system based on the life distributions of the components, subassemblies and/or assemblies ("black boxes") from which it is composed, as illustrated here.
©1999-2003. ReliaSoft Corp
Obtaining a system pdf from the pdfs of the components.
3
Basic System Reliability Terminology
• System
– A system is a collection of components, subsystems and/or assemblies arranged to a specific design in order to achieve desired functions with acceptable performance and reliability. The types of components, their quantities, their qualities and the manner in which they are arranged within the system have a direct effect on the system's reliability. – Often, the relationship between a system and its components is misunderstood or oversimplified. For example, the following statement is not valid: All of the components in a system have a 90% reliability at a given time, thus the reliability of the system is 90% for that time. Unfortunately, poor understanding of the relationship between a system and its constituent components can result in statements like this being accepted as factual, when in reality they are false.
7
©1999-2003. ReliaSoft Corp
Basic System Reliability Terminology
• Non-Repairable and Repairable Systems
– Systems can be generally classified into non-repairable and repairable systems. Non-repairable systems are those that do not get repaired when they fail. Specifically, the components of the system are not repaired or replaced when they fail. Most household products, for example, are non-repairable. This does not necessarily mean that they cannot be repaired, rather that it does not make economic sense to do so. For example, repairing a four-year-old microwave oven is economically unreasonable, since the repair would cost approximately as much as purchasing a new unit. – On the other hand, repairable systems are those that get repaired when they fail. This is done by repairing or replacing the failed components in the system. An automobile is an example of a repairable system. If the automobile is rendered inoperative when a component or subsystem fails, that component is typically repaired or replaced rather than purchasing a new automobile. In repairable systems, two types of distributions are considered: failure distributions and repair distributions. A failure distribution describes the time it takes for a component to fail. A repair distribution describes the time it takes to repair a component (timeto-repair instead of time-to-failure). In the case of repairable systems, the failure distribution itself is not a sufficient measure of system performance, since it does not account for the repair distribution. A new performance criterion called availability can be calculated, which accounts for both the failure and repair distributions.
4
©1999-2003. ReliaSoft Corp
Basic System Reliability Terminology
• Reliability Block Diagrams (RBDs)
– Block diagrams are widely used in engineering and science and exist in many different forms. They can also be used to describe the interrelation between the components and to define the system. When used in this fashion, the block diagram is then referred to as a reliability block diagram (RBD). – A reliability block diagram is a graphical representation of the components of the system and how they are reliability-wise related (connected). (Note: One can also think of an RBD as a logic diagram for the system based on its characteristics.) It should be noted that this may differ from how the components are physically connected. An RBD of a simplified computer system with a redundant fan configuration is shown in the following figure.
5
Basic System Reliabilitym Reliability Terminology
• The System Reliability Function
– After defining the properties of each block in a system, the blocks can then be connected in a reliability-wise manner to create a reliability block diagram for the system. The RBD provides a visual representation of the way the blocks are reliability-wise arranged. This means that a diagram will be created that represents the functioning state (i.e. success or failure) of the system in terms of the functioning states of its components. In other words, this diagram demonstrates the effect of the success or failure of a component on the success or failure of the system. For example, if all components in a system must succeed in order for the system to succeed, the components will be arranged reliability-wise in series. If one of two components must succeed in order for the system to succeed, those two components will be arranged reliability-wise in parallel. – The reliability-wise arrangement of components is directly related to the derived mathematical description of the system. The mathematical description of the system is the key to the determination of the reliability of the system. In fact, the system's reliability function is that mathematical description (obtained using probabilistic methods) and it defines the system reliability in terms of the component reliabilities. The result is an analytical expression that describes the reliability of the system as a function of time based on the reliability functions of its components.
Mathematics for Logisticians
Introduction and Overview
November 9, 2005
Kenneth East, C.P.L. Frank Vellella, C.P.L.
1
Mathematics in Logistics
• • • • • • • • • • • • • • • • • • Basics of Probability Measures of Central Tendency and Variability Data Description Probability Distributions Series and Parallel Component Reliability Failure Rates Probability of Success with Spares Availability Maintainability Factors Availability Inventory Management-Economic Order Quantity Forecasting-Moving Averages and Exponential Smoothing Regression and Correlation Analysis Queuing Theory Inflation, Discounting and Breakeven Analysis Decision Trees Linear Programming PERT/CPM Learning Curves
2
Basic System Reliability Terminology
The main objective of system reliability is the construction of a model (life distribution) that represents the times-to-failure of the entire system based on the life distributions of the components, subassemblies and/or assemblies ("black boxes") from which it is composed, as illustrated here.
©1999-2003. ReliaSoft Corp
Obtaining a system pdf from the pdfs of the components.
3
Basic System Reliability Terminology
• System
– A system is a collection of components, subsystems and/or assemblies arranged to a specific design in order to achieve desired functions with acceptable performance and reliability. The types of components, their quantities, their qualities and the manner in which they are arranged within the system have a direct effect on the system's reliability. – Often, the relationship between a system and its components is misunderstood or oversimplified. For example, the following statement is not valid: All of the components in a system have a 90% reliability at a given time, thus the reliability of the system is 90% for that time. Unfortunately, poor understanding of the relationship between a system and its constituent components can result in statements like this being accepted as factual, when in reality they are false.
7
©1999-2003. ReliaSoft Corp
Basic System Reliability Terminology
• Non-Repairable and Repairable Systems
– Systems can be generally classified into non-repairable and repairable systems. Non-repairable systems are those that do not get repaired when they fail. Specifically, the components of the system are not repaired or replaced when they fail. Most household products, for example, are non-repairable. This does not necessarily mean that they cannot be repaired, rather that it does not make economic sense to do so. For example, repairing a four-year-old microwave oven is economically unreasonable, since the repair would cost approximately as much as purchasing a new unit. – On the other hand, repairable systems are those that get repaired when they fail. This is done by repairing or replacing the failed components in the system. An automobile is an example of a repairable system. If the automobile is rendered inoperative when a component or subsystem fails, that component is typically repaired or replaced rather than purchasing a new automobile. In repairable systems, two types of distributions are considered: failure distributions and repair distributions. A failure distribution describes the time it takes for a component to fail. A repair distribution describes the time it takes to repair a component (timeto-repair instead of time-to-failure). In the case of repairable systems, the failure distribution itself is not a sufficient measure of system performance, since it does not account for the repair distribution. A new performance criterion called availability can be calculated, which accounts for both the failure and repair distributions.
4
©1999-2003. ReliaSoft Corp
Basic System Reliability Terminology
• Reliability Block Diagrams (RBDs)
– Block diagrams are widely used in engineering and science and exist in many different forms. They can also be used to describe the interrelation between the components and to define the system. When used in this fashion, the block diagram is then referred to as a reliability block diagram (RBD). – A reliability block diagram is a graphical representation of the components of the system and how they are reliability-wise related (connected). (Note: One can also think of an RBD as a logic diagram for the system based on its characteristics.) It should be noted that this may differ from how the components are physically connected. An RBD of a simplified computer system with a redundant fan configuration is shown in the following figure.