quiz2-2009

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Indian Institute of Technology, Bombay Chemical Engineering CL603: Optimization Quiz 2, Spring 2009 Follow  ALL  instr uctions careful ly: This is a  CLOSED BOOK exa m. Calculator s are permitted. Make reas onable assumptions and  CLEARLY indicate them in your answer book. Where possible, box your nal answers. Start each problem on a new sheet. 1. A thin-walled cylindrical press ure vessel with spherical ends is to be designed to minimize the total volume of the material used in its manufacture. The vessel must contain at least 25 m  of gas at a pres sure,    , of 0.35 MPa. The hoop stress,   , in the cylin der walls mu st not exc eed 200 MPa (       ). A sch ematic of the vessel is provided below. (a) Grap hicall y identify the feasib le region on an   vs.   plot. [2] (b) Using the Kuhn -T uck er conditions (i.e rst order condition s), determine the va lues of radiu s   and wall thickness   which mini mi ze the volume of the ma terial subject to the co ns traint s. [6] (c) Calculate the sensitivity of the optimal solution to small changes in the right hand sides of the inequality constraints. [2] Soln: 1. Let     . We want to minimize             where     m, subject to          MPa (        Mpa) and                  . The n                                                                 .          . The two cons train ts inte rsec t at        m. 2. KKT is    x      T  g   x   0    T g   x   0                                                                      Which results in 4 cases                       A:                  and            which implies     or        and     or     and hence no feasible solution exists.

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Indian Institute of Technology, Bombay Chemical Engineering

CL603: Optimization Quiz 2, Spring 2009

Follow   ALL   instructions carefully: This is a   CLOSED BOOK  exam. Calculators are permitted. Make reasonable

assumptions and  CLEARLY  indicate them in your answer book. Where possible, box your final answers. Start each

problem on a new sheet.

1. A thin-walled cylindrical pressure vessel with spherical ends is to be designed to minimize the total volume

of the material used in its manufacture. The vessel must contain at least 25 m     of gas at a pressure,     , of 

0.35 MPa. The hoop stress,  

 

  , in the cylinder walls must not exceed 200 MPa ( 

 

       ). A schematic

of the vessel is provided below.

(a) Graphically identify the feasible region on an 

  vs. 

  plot. [2]

(b) Using the Kuhn-Tucker conditions (i.e first order conditions), determine the values of radius 

  and

wall thickness 

  which minimize the volume of the material subject to the constraints. [6]

(c) Calculate the sensitivity of the optimal solution to small changes in the right hand sides of the

inequality constraints. [2]

Soln:

1. Let     

  . We want to minimize 

 

       

 

   

  where   

  m, subject to 

 

         

  MPa

(      

  Mpa) and 

 

       

 

   

 

       

   . Then

             

 

   

 

 

     

   

 

         

 

      

 

 

 

 

   

 

     

 

 

                       

 

           

  . The two constraints intersect at       

     m.

2. KKT is     

   

 

 

  T 

 x 

 

  0

 

 

 

Tg    x 

   0

 

 

   

 

       

 

 

   

 

       

 

 

 

   

 

 

   

 

     

 

 

   

 

   

 

 

     

 

       

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

     

 

   

 

 

 

 

   

   

   

Which results in 4 cases 

 

 

 

 

 

 

 

         

     

       

   

A:  

 

   

 

                   and                which implies      or            and      or

      and hence no feasible solution exists.

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B:  

 

   and hence                

 

               and                and        

 

       

 

     .

The last eq. gives     

  m while the second gives   

  or         

  and hence no feasible

solution again.

C: 

 

                   

 

     

  and                 

 

   

 

 

  and     

  . Taking ratio of 

first two eqs,                                 

  which gives         

  which is not feasible.

D:  

 

   

 

   .  

 

   gives        m.  

 

   gives          

   

  m.

               

 

       

 

             

                 

 

   

 

 

Second eq gives 

 

 

               

 

 

       

   

   

  and first gives

 

 

 

               

 

   

             

       

   

   

and hence we have a global minimum.

3.     

  x 

 

  d  

  at  d  

  is     

 

 

  . and hence the sensitivities are calculated already from above. The solutionis more sensitive to changes in the second constraint (volume) rather than the first (hoop stress).