# OPTIMIZATION Sayeed N Ghani PhD (Univ London), DIC (Imperial College), CEng (UK), MIEE (UK) Quality Six Sigma Green Belt Certified(USA) Copyright C 2007.

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OPTIMIZATION Sayeed N Ghani PhD (Univ London), DIC (Imperial College), CEng (UK), MIEE (UK) Quality Six Sigma Green Belt Certified(USA) Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

OPTIMIZATION 1 Copyright C
Index Systems Analysis vs.. Design Traditional Systems Design Optimum Systems Design Design Examples Example: Design of A Grain Silo First Design --- By Inexperienced Engineer Second Design --- By Experienced Engineer Third Design --- Optimum Design Rural Area --- Cost of Land Low First Pass --- Local Minimum Second Pass --- Global Minimum Urban Area --- Cost of Land High First Pass --- Local Minimum Second Pass --- Global Minimum Example: Design of an Electric Power Supply Pi-Section LC Filter First Design --- By Inexperienced Engineer Second Design --- By Experienced Engineer Third Design --- Optimum Design Minimum Cost Filter Design When Explicit Inequality Constraints Were Not Accounted For Comparison of Above Three Designs Maximization Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Systems Analysis vs.. Design
System Design Details Provided A Priori Analyze/ Calculate System Performance 1:1 Synthesize/ Design n:1 Many Systems Can Be Synthesized Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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 Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Systems Analysis vs. Design
System Design Details Provided A Priori Analyze/ Calculate System Performance 1:1 Synthesize/ Optimum Design 1:1 Best System Synthesized Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Optimum Systems Design (cont.)
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. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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). Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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.” Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Example: Design of a Grain Silo
h d All dimensions are in meters Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Volume V of the grain silo is: V = Pi/4 x d2 x h m3
Specification V = 200 m3 Substituting we obtain 200 = Pi/4 x d2 x h or d2 x h = m3 Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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 d2 x h = m3, and d = h d3 = or d = h = 6.34 m Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

In this example we want most economical design !
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. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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 manufacturer’s catalogues we derive the cost of a component to its size. We next obtain a best curve fit to obtain cost formulae.) Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Cost of base including land \$5000 + \$4000/m2
Optimum Design We perform a cost analysis and deduce the following econometric models. Cost of base including land \$ \$4000/m2 Cost of roof \$ \$1500/m2 Cost of silo walls \$ \$4000/m2 of wall area \$1000/m of height Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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) Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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 =< d <= infinity =< h <= infinity Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

and implicit constraint 254.655 =< d2h <= infinity
Defining design variables x1 = d and x2 = h the optimization model for the grain silo then becomes in generalized terms Minimize cost function F(x1, x2) = \$ \${420 x Pi/4 x (x1 + 4)2 }+ \$(40 x Pi x1 x2) + \$(1000 x2) Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

and implicit constraint 254.655 =< x12 x2 <= 99999
Subject to explicit constraints =< x1 <= =< x2 <= 99999 and implicit constraint =< x12 x2 <= 99999 Next a constrained optimizer ‘EVOP’ developed by this author was used to minimize the above objective function. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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 = d2h = m3 Volume = Pi/4 x d2h = 200 m3 Cost = \$55,643 Identical to Rule of Thumb Design by Experienced Engineer. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Second Pass and Global Minimum: Diameter d = 4.85 m Height h = 10.84 m
Implicit constraint XX = d2h = m3 Volume = Pi/4 x d2h = 200 m3 Cost = \$52,259 Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Saving over rule of thumb design = (\$ 55,643 - \$52,259)/ \$52,259
6.5 % 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). Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Design for Urban Area (cont.): First Pass and A Local Minimum:
Diameter d = 6.33 m Height h = 6.33 m Implicit constraint XX = d2h = m3 Volume = Pi/4 x d2h = 200 m3 Cost = \$402,841 Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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 = d2h = m3 Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Design for Urban Area (cont.): Volume = Pi/4 x d2h = 200 m3
Cost = \$240,828 Saving over rule of thumb design = (\$402,841 - \$240,828)/\$240,828 = 67.3 % For a silo of 500 m3 volume a saving of staggering 87 % has been achieved by EVOP. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

FINISHED FOR NOW FOLKS Copyright C

Design of An Electric Power Supply PI-Section L-C Filter
2nd Example Design of An Electric Power Supply PI-Section L-C Filter Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Example: Design a Power Supply L-C Filter
R = 1 K C2 C1 Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

To obtain a design we may choose C1 = C2 = 1 uF and obtain L = 346 H
Ripple factor r = 4.31 x 108/(f3 C1 C2 L R) Substituting r = 0.01, f = 50 and R = 1000 we obtain L = 346/(C1 C2) First design --- By Inexperienced Engineer To obtain a design we may choose C1 = C2 = 1 uF and obtain L = 346 H Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

We want most economical design !
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 ! Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Third Design --- Optimum Design
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 Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

have to bring, somehow, the concept of total system cost in our design formulae. This is how we do it. From the manufacturer’s 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. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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 Cents X X X X X X Henry uF Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

component size were determined to be
Cost of a capacitor = 5 + 1/uF cents Cost of an inductor = /H + 1/H cents Total system cost was then System Cost = Cost of C1 + Cost of C2 + Cost of L Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

System Cost = (5 + C1) + (5 + C2) + (50 + 5L + L2) 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. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

So, the mathematical model for optimization becomes
The constraint due to specification on ripple factor, as derived earlier, is L = 346/(C1 C2) So, the mathematical model for optimization becomes Minimize cost F(C1, C2, L) = 60 + (C C2) + L(5 + L) Subject to the following constraints on the design variables Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

0 < L < infinity 0 < C1 < infinity 0 < C2 < infinity
And implicit equality constraint L = 346/(C1 C2) This equality constraint is next introduced in the cost function F(C1, C2, L) to yield the cost function of reduced dimensionality Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

F(C1, C2) = 60 + (C1 + C2) + {346/(C1* C2)} * {5 + 346/(C1 + C2)}
We next account for non-negativity of our design variables C1 and C2 by introducing the following two transformations. x12 = C1 and x22 = C2 The cost function for unconstrained optimization now becomes Cost = F(x1, x2) = 60 + (x12 + x22) Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

+ {346/(x12 * x22)} * {5 + 346/(x12 * x22)} cents
x1 and x2 can now take negative values but capacitor values C1 and C2 will always remain positive. Next a versatile optimizer EVOP due to Ghani was used to minimize the above cost function. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

And implicit constraint: -99999 <= (x1 + x2) <= 99999
The problem presented to EVOP was Minimize F(x1, x2) = F(x1, x2) = 60 + (x12 + x22) + + {346/(x12 * x22)} * { /(x12 + x22)} cents Subject to explicit constraints: <= x1 <= <= x2 <= 99999 And implicit constraint: <= (x1 + x2) <= 99999 Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

EVOP calculated the following values of the objective function at the minimum:
X1* = 3.778; X2* = and F* = 99.92 Hence, C1* = = uF and C2* = = uF and L* = 346/(C1C2) = H Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Cost = 60 + (C1 + C2) + L(5 + L) cents we hand calculate the optimal
Using the relation Cost = 60 + (C1 + C2) + L(5 + L) cents we hand calculate the optimal Cost F* = 60 + ( ) ( ) = cents. Fletcher Powell’s famous gradient based variable metric quadratic convergent algorithm for unconstrained objective function yields the following optimum values. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

C1o = 14.4 uF, C2o = 14.4 uF, Cost Fo = 99.93 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 Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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: Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Minimum Cost Filter Design When Explicit Inequality Constraints Were Not Accounted For
C1 = x 104 F C2 = x 104 F L = uH Cost = Mega Dollars Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

AND wouldn’t all power electronics engineers like myself !!
Only if I could make a negative capacitor capable of carrying currents at power level, wouldn’t I have been a double billionaire by just making (not even selling) only 1 number of such a power supply !! AND wouldn’t all power electronics engineers like myself !! Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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 = \$ Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

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. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

Minimize -Fo (x1, x2, …, xn) will yield: Maximize Fo (x1, x2, …, xn)
Maximization Finally, if maximization is needed (instead of minimization), then negate the objective function and then minimize as usual. Minimize -Fo (x1, x2, …, xn) will yield: Maximize Fo (x1, x2, …, xn) Constraints should be kept unchanged. Copyright C 2007 by Sayeed Nurul Ghani. All rights reserved.

FINISHED FOR NOW FOLKS Copyright C