know the sensitivity of the optimal solution but do not have the original optimization program or do not want to rework the problem. If the sensitivity information is obtained simultaneously with the optimal solution, the users of the results can quickly locate the new optimal policy corresponding to new values of the parameters. I n some cases, the sensitivity information is obtained as soon as the optimization problem is solved without additional effort-for example, the sensitivity results in case C were obtained a t the same time the problem was solved.
parameter vector
W
=
X
= state vector; mass fraction of B = adjoint vector
Y, z
GREEKLETTERS = initial value vector = increment = decision vector; temperature = optimal value of 0 e 9 = Lagrangian function h = Lagrange multiplier ic = constraint vector U = sensitivity
a A 9
Acknowledgment
The authors are indebted to Rutherford Ark, University of Minnesota, for calling to our attention the adjointness in the sensitivity equations. The financial support of the Office of Coal Research, Department of the Interior, Washington, D. C., is acknowledged. Nomenclature A, B = chemical species = constant in objective function C El, Ea = activation energy, B.t.u./lb.-mole fi = performance function defined in Equation 5 P = function defined in Equation 20 h = function defined in Equation 16 H = Hamiltonian function I = unit matrix J = objective function j = modified objective function = reaction rate constant, hr.-l k l , kz k l o , k 2 0 = frequency factor, hr.-I L = axial position in tubular reactor L‘ = length of tubular reactor, ft. m = dimension of constraint vector on decisions = dimension of parameter vector, w P = dimension of equality constraint vector 4 r = dimension of decision vector 0 R = gas constant, B.t.u./(lb.-mole) (” R.) = dimension of state vector, x S = sensitivity of J to change in w S,J t = time; holding time T = final value of time; total holding time; hr. U = linear velocity, ft./hr. U = X(@L/&); adjoint vector
literature Cited Aris, R., “The Optimal Design of Chemical Reactors,” Academic Press, New York, 1961. Bode, H. W., “Network Analysis and Feedback Amplifier Design,” Chap. 4, 5, Van Nostrand, New York, 1945. Boot, J. C. G., Operations Res. 11, 771 (1963). Boot, J. C. G., “Quadratic Programming,” Rand McNally, Chicago, 1964. Bryson, A. E., Jr., Denham, W.F., Dreyfus, S. E., AZAA J. 1, 2544 (1963). Chang, S. S. L., “Synthesis of Optimum Control Systems,” Chap. 8, McGraw-Hill, New York, 1961. Chang, T. M., “Sensitivity Analysis in Optimum Process Design,” Ph.D. Dissertation, West Virginia University, Morgantown, LV. Va., 1967. Dorato, P., ZEEE Trans. Automatic Control AC-8, 256 (1963). Fan, L. T., “The Continuous Maximum Principle,” Wiley, New York, 1966. Fan, L. T., Wang, S. C., “The Discrete Maximum Principle,” LYiley, New York, 1964. Garvin, W.W., “Introduction to Linear Programming,” Chap. 4, McGraw-Hill. New York. 1960. Hadley, G., “Linear Programming,” Chap. 11, Addison-Wesley, Reading, Mass., 1962. Horn, F., Jackson, R., IND.ENG.CHEM.FUNDAMENTALS 4, 110 (,-,--,. 1 Oh
