Presentation is loading. Please wait.

Presentation is loading. Please wait.

1 OPTIMIZATION Sayeed N Ghani PhD (Univ London), DIC (Imperial College), CEng (UK), MIEE (UK) Quality Six Sigma Green Belt Certified(USA) Copyright2007.

Similar presentations


Presentation on theme: "1 OPTIMIZATION Sayeed N Ghani PhD (Univ London), DIC (Imperial College), CEng (UK), MIEE (UK) Quality Six Sigma Green Belt Certified(USA) Copyright2007."— Presentation transcript:

1 1 OPTIMIZATION Sayeed N Ghani PhD (Univ London), DIC (Imperial College), CEng (UK), MIEE (UK) Quality Six Sigma Green Belt Certified(USA) Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

2 2 OPTIMIZATION 1 Index 2 Systems Analysis vs.. Design 3 Traditional Systems Design 4 Optimum Systems Design 6 Design Examples Example: Design of A Grain Silo 13 First Design --- By Inexperienced Engineer 15 Second Design --- By Experienced Engineer 17 Third Design --- Optimum Design 20 Rural Area --- Cost of Land Low 20 First Pass --- Local Minimum 25 Second Pass --- Global Minimum 26 Urban Area --- Cost of Land High 27 First Pass --- Local Minimum 28 Second Pass --- Global Minimum 29 Example: Design of an Electric Power Supply Pi-Section LC Filter 32 First Design --- By Inexperienced Engineer 34 Second Design --- By Experienced Engineer 35 Third Design --- Optimum Design 37 Minimum Cost Filter Design When Explicit Inequality 51 Constraints Were Not Accounted For Comparison of Above Three Designs 53 Maximization 55 Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

3 3 Systems Analysis vs.. Design Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C System Design Details Provided A Priori System Performance Analyze/ Calculate 1:1 Many Systems Can Be Synthesized Synthesize/ Design n:1

4 4 Traditional Systems Design In traditional systems design the objective is merely to meet the specifications. There is no formal attempt to reach the best design in the strict mathematical sense of minimizing cost or weight or volume or maximizing profit. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

5 5 Traditional Systems Design (cont..) Further in traditional systems design a highly skilled engineer is in the design loop making sound engineering decisions at every stage of the design process. The process undergoes many manual iterations before the design can be Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

6 6 Optimum Systems Design finalized making it a slow and very costly process. The science of optimization is a formalism that allows not only all specifications (design constraints) to be met, but would also yield design which is the best in terms of some figure(s) of merit. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

7 7 Systems Analysis vs. Design Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C System Design Details Provided A Priori System Performance Analyze/ Calculate 1:1 Best System Synthesized Synthesize/ Optimum Design 1:1

8 8 Optimum Systems Design (cont.) Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C It is a completely automated process that allows lesser skilled and experienced engineers to create optimum design. Optimization is applicable to all numerate disciplines including fuzzy systems.

9 9 Optimum Systems Design (cont.) Fuzzy logic, fuzzy sets, fuzzy relations and fuzzy reasoning allows synthesis of high performance control systems that would beat, hands down, any linear counterpart (if properly designed). Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

10 10 Optimum Systems Design (cont.) Fuzzy concept is ideally suited to model poorly defined processes that could only be described in qualitative terms via linguistic variables. If temperature is extremely low then set fuel injection high. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

11 11 Optimum Systems Design (cont.) Objective functions involving fuzzy systems are known to possess multiple minima and difficult to optimize. Optimization is one of the core concepts used in DFSS (Design for Six Sigma) for product to service and for manufacturing to transaction. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

12 12 Optimum Systems Design (cont.) Optimization is the only way to ensure that an enterprise not only meets all its requirements but is also the best in its widest possible meaning. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

13 13 Example: Design of a Grain Silo d h d + 4 All dimensions are in meters Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

14 14 Volume V of the grain silo is: V = Pi/4 x d 2 x h m 3 Specification V = 200 m 3 Substituting we obtain 200 = Pi/4 x d 2 x h or d 2 x h = m 3 Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

15 15 First design --- By Inexperienced Engineer Arbitrarily choose d = 1 m Therefore, h = m The design will of course work, but the problem is that it will be too expensive. What we have actually done really is to apply rule of thumb saying that we shall arbitrarily choose diameter d = 1 m. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

16 16 This applies an artificial constraint on our design variable d to yield a unique solution. Now in practice rules of thumb used by competent engineers are never so unrealistic. But nevertheless they are only guess work. If we study a number of grain silo designs we will repeatedly come across designs more or less of square shape. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

17 17 Second design --- By Experienced Engineer The engineer has been making similar grain silos for many years. From past experience he bases his design on roughly a square shape. With d 2 x h = m 3, and d = h d 3 = or d = h = 6.34 m Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

18 18 But he has not attempted, in any way, to obtain the best design in terms of some figure of merit which is cost in this case. In this example we want most economical design ! So we see traditional design approach (use of rules of thumb) has no capability to satisfy our quest for the best. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

19 19 Since we are seeking the most cost effective design we shall have to bring, somehow, the concept of total system cost in our design formulae. (In general terms this is how we do it. From the manufacturers catalogues we derive the cost of a component to its size. We next obtain a best curve fit to obtain cost formulae.) Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

20 20 Optimum Design We perform a cost analysis and deduce the following econometric models. Cost of base including land $ $4000/m 2 Cost of roof $ $1500/m 2 Cost of silo walls $ $4000/m 2 of wall area + $1000/m of height Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

21 21 Total cost = Cost of base + Cost of roof + Cost of silo wall. Cost of base including land $ $400 x Pi/4 x (d + 4) 2 Cost of roof $ $20 x Pi/4 x (d + 4) 2 Cost of silo wall $ $40 x Pi x d x h + $1000h Total cost of the silo = $ ${420 x Pi/4 x (d + 4) 2 }+ $(40 x Pi x d x h) + $(1000 x h) Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

22 22 The optimization model (cost function for constrained optimization) for the grain silo is then Minimize cost function F(d, h) = $ ${420 x Pi/4 x (d + 4) 2 }+ $(40 x Pi x d x h) + $(1000 x h) Subject to explicit constraints 0 =< d <= infinity 0 =< h <= infinity Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

23 23 and implicit constraint =< d 2 h <= infinity Defining design variables x 1 = d and x 2 = h the optimization model for the grain silo then becomes in generalized terms Minimize cost function F(x 1, x 2 ) = $ ${420 x Pi/4 x (x 1 + 4) 2 }+ $(40 x Pi x 1 x 2 ) + $(1000 x 2 ) Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

24 24 Subject to explicit constraints 0 =< x 1 <= =< x 2 <= and implicit constraint =< x 1 2 x 2 <= Next a constrained optimizer EVOP developed by this author was used to minimize the above objective function. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

25 25 And here are the results of optimization from subroutine EVOP. First Pass and A Local Minimum: Diameter d = 6.34 m Height h = 6.34 m Implicit constraint XX = d 2 h = m 3 Volume = Pi/4 x d 2 h = 200 m 3 Cost = $55,643 Identical to Rule of Thumb Design by Experienced Engineer. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

26 26 Second Pass and Global Minimum: Diameter d = 4.85 m Height h = m Implicit constraint XX = d 2 h = m 3 Volume = Pi/4 x d 2 h = 200 m 3 Cost = $52,259 Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

27 27 Saving over rule of thumb design = ($ 55,643 - $52,259)/ $52, % only. Design for Urban Area With Ten Fold Increase in Cost of Land: Base = $50,000 + ${4000 x Pi/4 * (d +4) 2 }. Cost of roof and wall remains unchanged (Page 21). Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

28 28 Design for Urban Area (cont.): First Pass and A Local Minimum: Diameter d = 6.33 m Height h = 6.33 m Implicit constraint XX = d 2 h = m 3 Volume = Pi/4 x d 2 h = 200 m 3 Cost = $402,841 Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

29 29 Design for Urban Area (cont.): Identical to Rule of Thumb Design by Experienced Engineer. Second Pass and Global Minimum: Diameter d = 2.44 m Height h = m Implicit constraint XX = d 2 h = m 3 Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

30 30 Design for Urban Area (cont.): Volume = Pi/4 x d 2 h = 200 m 3 Cost = $240,828 Saving over rule of thumb design = ($402,841 - $240,828)/$240,828 = 67.3 % For a silo of 500 m 3 volume a saving of staggering 87 % has been achieved by EVOP. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

31 31 FINISHED FOR NOW FOLKS Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

32 32 2 nd Example Design of An Electric Power Supply PI-Section L-C Filter Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

33 33 Example: Design a Power Supply L-C Filter R = 1 K L C1C1 C2C2 Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

34 34 Ripple factor r = 4.31 x 10 8 /(f 3 C 1 C 2 L R) Substituting r = 0.01, f = 50 and R = 1000 we obtain L = 346/(C 1 C 2 ) First design --- By Inexperienced Engineer To obtain a design we may choose C 1 = C 2 = 1 uF and obtain L = 346 H Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

35 35 Second design --- By Experienced Engineer Now in practice rules of thumb used by competent engineers are never so unrealistic. But nevertheless they are only guess work. If we study a number of power supply designs we will repeatedly come across figures like 32 uF. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

36 36 This will yield L = 338 mH – a more realistic value to use for the inductor. But we have not attempted, in any way, to obtain the best design in terms of some figure of merit which is cost in this case. We want most economical design ! Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

37 37 So we see traditional design approach (use of rules of thumb) has no capability to satisfy our quest for the best. Third Design --- Optimum Design Since we are seeking the most cost effective design we shall Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

38 38 have to bring, somehow, the concept of total system cost in our design formulae. This is how we do it. From the manufacturers catalogues we derive the cost of a component to its size. We next obtain a best curve fit to obtain a mathematical expression relating cost to size. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

39 39 This we can do either on a computer, or manually on a graph paper as shown below. From the two figures below the functional relationship between cost and Cents uF Henry X X X X X X Cents Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

40 40 component size were determined to be Cost of a capacitor = 5 + 1/uF cents Cost of an inductor = /H + 1/H 2 cents Total system cost was then System Cost = Cost of C 1 + Cost of C 2 + Cost of L Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

41 41 System Cost = (5 + C 1 ) + (5 + C 2 ) + (50 + 5L + L 2 ) cents = 60 + (C 1 + C 2 ) + L(5 + L) cents Having obtained the total system cost, we have to next introduce the constraints on the design variables imposed by the specification on the ripple factor in our optimization model. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

42 42 The constraint due to specification on ripple factor, as derived earlier, is L = 346/(C 1 C 2 ) So, the mathematical model for optimization becomes Minimize cost F(C 1, C 2, L) = 60 + (C 1 + C 2 ) + L(5 + L) Subject to the following constraints on the design variables Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

43 43 0 < L < infinity 0 < C 1 < infinity 0 < C 2 < infinity And implicit equality constraint L = 346/(C 1 C 2 ) This equality constraint is next introduced in the cost function F(C 1, C 2, L) to yield the cost function of reduced dimensionality Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

44 44 F(C 1, C 2 ) = 60 + (C 1 + C 2 ) + {346/(C 1 * C 2 )} * { /(C 1 + C 2 )} We next account for non-negativity of our design variables C 1 and C 2 by introducing the following two transformations. x 1 2 = C 1 and x 2 2 = C 2 The cost function for unconstrained optimization now becomes Cost = F(x 1, x 2 ) = 60 + (x x 2 2 ) Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

45 45 + {346/(x 1 2 * x 2 2 )} * { /(x 1 2 * x 2 2 )} cents x 1 and x 2 can now take negative values but capacitor values C 1 and C 2 will always remain positive. Next a versatile optimizer EVOP due to Ghani was used to minimize the above cost function. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

46 46 The problem presented to EVOP was Minimize F(x 1, x 2 ) = F(x 1, x 2 ) = 60 + (x x 2 2 ) + + {346/(x 1 2 * x 2 2 )} * { /(x x 2 2 )} cents Subject to explicit constraints: <= x 1 <= <= x 2 <= And implicit constraint: <= (x 1 + x 2 ) <= Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

47 47 EVOP calculated the following values of the objective function at the minimum: X 1 * = 3.778; X 2 * = and F* = Hence, C 1 * = = uF and C 2 * = = uF and L * = 346/(C 1 C 2 ) = H Copyright 2007 by Sayeed Nurul Ghani. All rights reserved. C

48 48 Using the relation Cost = 60 + (C 1 + C 2 ) + L(5 + L) cents we hand calculate the optimal Cost F * = 60 + ( ) ( ) = cents. Fletcher Powells famous gradient based variable metric quadratic convergent algorithm for unconstrained objective function yields the following optimum values. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

49 49 C 1 o = 14.4 uF, C 2 o = 14.4 uF, Cost F o = cents. The value of L was rounded up to 1.7 H. Minimum Cost Filter Design C1 = 14.4 uF C2 = 14.4 uF L = 1.7 H Cost = cents Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

50 50 If explicit inequality constraints on the design variables were not accounted for, the computer being nothing more than a fast number crunching idiot will churn out an impossible, unrealistic design as follows: Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

51 51 Minimum Cost Filter Design When Explicit Inequality Constraints Were Not Accounted For C 1 = x 104 F C 2 = x 104 F L = uH Cost = Mega Dollars Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

52 52 Only if I could make a negative capacitor capable of carrying currents at power level, wouldnt I have been a double billionaire by just making (not even selling) only 1 number of such a power supply !! AND wouldnt all power electronics engineers like myself !! Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

53 53 Cost Comparison Finally, let us make a cost comparison of all three designs. Designed by popular vote by students. C1 = C2 = 1 uf L = 346 H Cost = $ Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

54 54 Designed by experienced engineer using rule of thumb. C1 = C2 = 32 uf L = H Cost = $ 1.26 Optimized design is $1 only. (26 % cheaper and no skilled designer overhead) Without design experience optimization is indispensable tool. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

55 55 Maximization Finally, if maximization is needed (instead of minimization), then negate the objective function and then minimize as usual. Minimize -F o (x 1, x 2, …, x n ) will yield: Maximize F o (x 1, x 2, …, x n ) Constraints should be kept unchanged. Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C

56 56 FINISHED FOR NOW FOLKS Copyright2007 by Sayeed Nurul Ghani. All rights reserved. C


Download ppt "1 OPTIMIZATION Sayeed N Ghani PhD (Univ London), DIC (Imperial College), CEng (UK), MIEE (UK) Quality Six Sigma Green Belt Certified(USA) Copyright2007."

Similar presentations


Ads by Google