Engineering Optimization: Theory and Practice, Fourth Edition

574 Dynamic Programming

periodt 1 tot 2 is given by

f=

∫t 2

t 1

[

p

(

a+

b

p

)

−c

(

p,

dp

dt

, t

)

−x(t)

]

dt (E 1 )

wherep=p{x(t), t}. Thus the optimization problem can be stated as follows: Find

x(t),t 1 ≤t≤t 2 , which maximizes the total profit,fgiven by Eq. (E 1 ).

Example 9.7 Consider the problem of determining the optimal temperature distribu-

tion in a plug-flow tubular reactor [9.1]. Let the reactions carried in this type of reactor

be shown as follows:

X 1

k 1

⇄

k 2

X 2

k 3

−−−→X 3

whereX 1 is the reactant,X 2 the desired product, andX 3 the undesired product, and

k 1 ,k 2 , andk 3 are called rate constants. Letx 1 andx 2 denote the concentrations of the

productsX 1 andX 2 , respectively. The equations governing the rate of change ofthe

concentrations can be expressed as

dx 1

dy

+k 1 x 1 =k 2 x 2 (E 1 )

dx 2

dy

+k 2 x 2 +k 3 x 2 =k 1 x 1 (E 2 )

with the initial conditionsx 1 (y= 0 )=c 1 andx 2 (y= 0 )=c 2 , where yis the normal-

ized reactor length such that 0≤y≤1. In general, the rate constants depend on the

temperature (t) and are given by

ki=aie− b(i/t), i= 1 , 2 , 3 (E 3 )

whereaiandbiare constants.

If the objective is to determine the temperature distributiont (y), 0≤y≤1, to

maximize the yield of the productX 2 , the optimization problem can be stated as

follows:

Findt (y), 0≤y≤1, which maximizes

x 2 ( 1 )−x 2 ( 0 )=

∫ 1

y= 0

dx 2 =

∫ 1

0

(k 1 x 1 −k 2 x 2 −k 3 x 2 ) dy

wherex 1 (y) andx 2 (y) have to satisfy Eqs. (E 1 ) nd (Ea 2 ) Here it is assumed that the.

desired temperature can be produced by some external heating device.

The classical method of approach to continuous decision problems is by the calcu-

lus of variations.†However, the analytical solutions, using calculus of variations, cannot

beobtained except for very simple problems. The dynamic programming approach, on

the other hand, provides a very efficient numerical approximation procedure for solving

continuous decision problems. To illustrate the application of dynamic programming

†See Section 12.2 for additional examples of continuous decision problems and the solution techniques using

calculus of variations.