- 2. 3.1 Introduction (2) Examples of axioms in Geometry: Points and lines are names for the elements of two (distinct) sets. Incidence is a relationship that may (or may not) hold between a particular point and a particular line. The followings are examples of axioms: 1) For every two points, there exists a line incident with both points. 2) For every two points, there is no more than one line incident with both points. 3) There exist at least two points incident with each line. 4) There exist at least three points. Not all points are incident with the same line. Examples of axioms in Physics: 1) Newton’s law: F = ma 2) Thermodynamic principles 3) ……
- 3. Axiomatic design is a design methodology that was created and popularized by Professor Suh of MIT (used be the President of KAIST). It is a design framework that is feasible on all design disciplines. The Independence Axiom, The Information Axiom An axiom is a statement accepted without proof as an underlying assumption of a formal mathematical theory. It cannot be proved. If a counter example is found for an axiom, the axiom becomes obsolete. Geometry, Laws in Physics, Thermodynamic principles 3.1 Introduction
- 4. Design Axioms Axiom 1: The Independence Axiom Maintain the independence of FRs. Alternate Statement 1: An optimal design always maintains the independence of FRs. Alternate Statement 2: In an acceptable design, DPs and FRs are related in such a way that a specific DP can be adjusted to satisfy its corresponding FR without affecting other functional requirements. Axiom 2: The Information Axiom Minimize the information content of the design. Alternate Statement: The best design is a functionally uncoupled design that has minimum information content. 3.2. Design Axioms
- 5. Usage of the axioms: Analysis of design Find designs that satisfy the Independence Axiom. Determine the final design. Is the no. of designs sufficient? Find the best design with the Information Axiom. Multiple designs? No Yes Yes No 3.2. Design Axioms (2) Figure 3.1. Flow chart of the application of axiomatic design
- 6. The Independence Axiom: The FR-DP relationship should be independent. 3 2 1 3 2 1 , DP DP DP FR FR FR DP FR equation design : ADP FR 3 2 1 33 32 31 23 22 21 13 12 11 3 2 1 DP DP DP A A A A A A A A A FR FR FR FR: FR vector, DP: DP vector, A: design matrix 3.3 Independence Axiom
- 7. 3 2 1 33 22 11 3 2 1 0 0 0 0 0 0 DP DP DP A A A FR FR FR 1 11 1 DP A FR 2 22 2 DP A FR 3 33 3 DP A FR 3 2 1 33 32 31 22 21 11 3 2 1 0 0 0 DP DP DP A A A A A A FR FR FR 1 11 1 DP A FR 2 22 1 21 2 DP A DP A FR 3 33 2 32 1 31 3 DP A DP A DP A FR 3.3 Independence Axiom (2) 1) Uncoupled design: Each DP satisfies the corresponding FR independently. - diagonal matrix 2) Decoupled design: The Independence Axiom is satisfied when the design sequence is right. - triangular matrix
- 8. (3) Coupled design: No sequences of DPs can satisfy the FRs independently. –general matrix 3 2 1 33 32 31 23 22 21 13 12 11 3 2 1 DP DP DP A A A A A A A A A FR FR FR 3 13 2 12 1 11 1 DP A DP A DP A FR 3 23 2 22 1 21 2 DP A DP A DP A FR 3 33 2 32 1 31 3 DP A DP A DP A FR Constraints (Cs): Cs can be defined regardless of the independence of the FR-DP relationship. 3.3 Independence Axiom (3)
- 9. (a) Vertically hung door (b) Horizontally hung door Example 3.1 [Design of a Refrigerator Door] FRs for a refrigerator door are as follows: FR1: Provide access to the items stored in the refrigerator. FR2: Minimize energy loss. Which door is better between the following doors? 3.4 Application of the Independence Axiom Figure 3.2. Refrigerator doors
- 10. 2 1 2 1 0 DP DP X X X FR FR 2 1 2 1 0 0 DP DP X X FR FR X: nonzero value (relationship) Example 3.1 [Design of a Refrigerator Door] Vertically hung door in Figure 3.2(a) DP1: Vertically hung door DP2: Thermal insulation in the door (2) Horizontally hung door in Figure 3.2(b) DP1: Horizontally hung door DP2: Thermal insulation in the door Which one is better? 3.4 Application of the Independence Axiom (2)
- 11. Example 3.2 [Design of a Water Faucet] Some commercial water faucets are evaluated. FRs for the faucet are defined as follows: FR1: Control the flow of water (Q). FR2: Control the temperature of water (T). 3.4 Application of the Independence Axiom (3)
- 12. Example 3.2 [Design of a Water Faucet] For the one in Figure 3.3(a) DP1: Angle DP2: Angle 1 2 ) ( ) ( ) ( ) ( 2 2 1 1 2 1 DP DP X X X X T FR Q FR 1 2 3.4 Application of the Independence Axiom (4) Figure 3.3.(a) Coupled design
- 13. Example 3.2 [Design of a Water Faucet] For the one in Figure 3.3(b) DP1: Angle DP2: Angle 1 2 ) ( ) ( 0 0 ) ( ) ( 2 2 1 1 2 1 DP DP X X T FR Q FR Cold water Hot water 2 1 3.4 Application of the Independence Axiom (5) Figure 3.3.(b) Coupled design
- 14. ) ( ) ( 0 0 ) ( ) ( 2 1 2 1 DP Y DP X X T FR Q FR Which one is the best? Is it different from what you expected? Y Example 3.2 [Design of a Water Faucet] For the one in Figure 3.3(c) DP1: Displacement Y DP2: Angle 3.4 Application of the Independence Axiom (6) Figure 3.3.(c) Coupled design
- 15. 3 2 1 3 2 1 0 0 0 0 DP DP DP X X x X X FR FR FR 3 2 1 3 2 1 0 0 0 0 DP DP DP X X x X X FR FR FR for Figure 2.1(a) Example 3.3 [Axiomatic Design of the Toaster in Example 2.1] Make the design matrices for the products in Figure 2.1(a) and Figure 2.1(b) Solution 3.4 Application of the Independence Axiom (7) for Figure 2.1(b)
- 16. (a) Functional domain FR FR1 FR2 FR11 FR12 … … FR21 FR22 … DP DP1 DP2 DP11 DP12 … … DP21 DP22 … ① ② ③ ④ (b) Physical domain For complicated systems: we need a decomposition which yields a hierarchy. The zigzagging process 3.4 Application of the Independence Axiom (8) Figure 3.4. Zigzagging process between domains
- 17. FR: 서울에서 부산으로 가라. 지그재그과정의 예 (서울에서 부산가는 법) 서울에서 부산으로 가라. 비행기 비행장으로 가라. 표를 사라. … 비행기를 타라. 서울에서 부산으로 가라. 기차 기차역으로 가라. 표를 사라. … 기차를 타라. (1) (2) 상위단계의 DP가 하위단계의 FR을 결정한다.
- 18. Example 3.4 [Decomposition of Example 2.2] The top level (first level) FRs and DPs are: FR1: Freeze food or water for long-term preservation. FR2: Maintain food at a cold temperature for short-term preservation. DP1: The freezer section DP2: The chiller section 3.4 Application of the Independence Axiom (9)
- 19. Example 3.4 [Decomposition of Example 2.2] The second (first level) FRs are: FR11: Maintain the temperature of the freezer section in the range of . FR12: Maintain a uniform temperature in the freezer section. FR13: Control the relative humidity to 50% in the freezer section. FR21: Maintain the temperature of the chiller section in the range of . FR22: Maintain a uniform temperature in the chiller section within of the preset temperature . C 2 C 18 C 3 C 2 C 5 . 0 3.4 Application of the Independence Axiom (10)
- 20. 13 11 12 13 11 12 0 0 0 0 DP DP DP X X X X X FR FR FR Example 3.4 [Decomposition of Example 2.2] The second level DPs are: DP11: Sensor/compressor system that activates the compressor when the temperature of the freezer section is different from the preset one. DP12: Air circulation system that blows the air into the freezer and circulates it uniformly. DP13: Condenser that condenses the moisture in the returned air when the dew point is exceeded. 3.4 Application of the Independence Axiom (11)
- 21. 21 22 21 22 0 DP DP X X X FR FR Example 3.4 [Decomposition of Example 2.2] The second level DPs are: DP21: Sensor/compressor system that activates the compressor when the temperature of the chiller section is different from the preset one. DP22: Air circulation system that blows the air into the chiller section and circulates it uniformly. 3.4 Application of the Independence Axiom (12)
- 22. 21 22 13 11 12 21 22 13 11 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 DP DP DP DP DP X X X X X X X X FR FR FR FR FR Example 3.4 [Decomposition of Example 2.2] The entire design equation is 3.4 Application of the Independence Axiom (13)
- 23. Physical integration There is a saying that a simple design is a good one. A good design makes one DP satisfy multiple FRs? A coupled design is better? This is the case where multiple DPs make a physical entity. Multiple DPs satisfy FRs of the same number. Physical Integration: recommended 3.4 Application of the Independence Axiom (14)
- 24. 2 1 2 1 0 0 DP DP X X FR FR Example 3.5 [Bottle-can opener] Given FRs are FR1: Design a device that can open bottles. FR2: Design a device that can open cans. 3.4 Application of the Independence Axiom (15) DP2 DP1
- 25. 12 FRs and 12 DPs: decoupled design Example 3.6 [Beverage Can Design] Another example for physical integration 3.4 Application of the Independence Axiom (16)
- 26. In the designing process, the Independence Axiom should be satisfied first. When multiple designs that satisfy the Independence Axiom are found, the Information axiom is utilized to find the best design. The best design is the one with minimum information. Generally, the information is related to complexity. How can we measure the complexity? How can we quantify the information content? 3.5 The Information Axiom
- 27. p I / 1 log2 The information can be defined in various ways. Up to now, one measure is used for the information content. where I is the information content and p is the probability of success to satisfy an FR with a DP. The reciprocal of p is used to make the larger probability have less information. The logarithm function is used to enhance additivity. The base of the logarithm is 2 to express the information content with the bit unit. 3.5 The Information Axiom (2)
- 28. 3 2 1 33 22 11 3 2 1 0 0 0 0 0 0 DP DP DP A A A FR FR FR Suppose p1, p2 and p3 are probabilities of satisfying FR1, FR2 and FR3 with DP1, DP2 and DP3, respectively. The total information Itotal is 3 1 2 3 1 total 1 log i i i i p I I For the following uncoupled design: 3.5 The Information Axiom (3)
- 29. If p1 is the probability that DP1 satisfies FR1, then the probability that DP2 satisfies FR2 under the satisfaction of FR1 by DP1 is a conditional probability. Suppose it is p21. Then the probability of success p that both FR1 and FR2 are satisfied is p=p1p21. The total information content for p is 2 1 2 1 0 DP DP X X X FR FR 2 1 21 2 1 2 21 1 2 2 log log ) ( log log I I p p p p p I For the following decoupled design: 3.5 The Information Axiom (4)
- 30. System range (Asr): response (output) Design range: target, Common range: Acr sr cr / A A ps ) / ( log sr cr 2 A A I Design range Probability density Probability density function of the system Common range FR 1 3 5 7 9 Information content can be calculated by using the probability density function in the following figure: 3.5 The Information Axiom (5)
- 31. Example 3.7 [An Example of Calculating Information Content] For the bottle-can opener problem The probability of satisfying FR1 with DP1: 0.9 The probability of satisfying FR2 with DP2: 0.85 The total information content is bits) ( 3865 . 0 2345 . 0 1520 . 0 85 . 0 1 log 9 . 0 1 log 2 2 2 1 total I I I What if we do not have physical integration? 3.6 Application of the Information Axiom
- 32. city A city B Price $45,000 - $60,000 $70,000 - $90,000 Commuting time 35-50 min 20-30 min Example 3.9 [Calculation of the Information Content Using the Probability Density Function] A person defines two functional requirements to buy a house as follows: FR1: Let the price range be from $50,000 to $80,000. FR2: Let the commuting time be within 40 minutes. The following table shows the conditions of city A and city B. 3.6 Application of the Information Axiom (2)
- 33. For city A 59 . 1 5 15 log , 59 . 0 1 5 . 1 log 2 2 2 1 A A I I ) bits ( 18 . 2 2 1 A A A I I I 0 . 1 1 2 log2 1 B I 0 . 0 10 10 log2 2 B I bits) ( 0 . 1 2 1 B B B I I I Which one is better? Design range Bias Probability density FR Target Probability density function of the system Common range Variation from the peak value Example 3.9 [Calculation of the Information Content Using the Probability Density Function] For city B 3.6 Application of the Information Axiom (3)
- 34. Time Two axioms are independent of each other. The flow in Figure 3.1 is recommended. The numbers of FRs and DPs should be the same (ideal design). The no. of DPs is smaller: New DPs should be added. The no. of DPs is larger (redundant design): Some DPs are eliminated or specific DPs are fixed. Axiomatic design is useful in conceptual design. It can be used for creative design or evaluating an existing design. 3.7 Discussion Solution neutral environment