For this case either the cost or profit can be represented by a polynomial. The same procedure employing classical methods (8) will be used to obtain the dual problem, and the techniques will be essentially the same to find the optimum. However, the main difference is that stationary points will be found, and there will be no guarantee that either a maximum or a minimum has been located. It will be necessary to use the methods of Chapter 2 or local exploration to determine their character.
It is convenient to group the positive terms and the negative terms together to represent a general polynomial. This is written as:
An example of this equation, given below, will be used to illustrate the solution technique for polynomials.
and comparing equations (3-17) and (3-18) gives:
for i = 1, 2, 3, ..., N, which, after multiplying by xi /y can be written as:
for i = 1, 2, ..., N.
The definition of the optimal weights (equation 3-7) can now be used to give the orthogonality conditions for polynomial optimization, i.e.:
for n = 1, 2, ..., N, where the subscript n has been used in place of i for convenience.
Also the normality condition is obtained the same way as equation (3-6) by dividing equation (3-17) by the optimal value of y, which is known in principle from the solution of the set of equations given by equatio (3-20). The result is:
The only algebraic manipulations that remain are to obtain the equation comparable to equation (3-14) for a polynomial profit or cost function. The procedure is the same and uses equation (3-22) as follows:
Again the definition of the optimal weights, equation (3-7), is used to eliminate y from the right-hand side of equation (3-23) and introduce ct and wt as:
and the above can be written as
The term in the second bracket can be written as
Using equation (3-21) and performing the manipulations done to obtain equation (3-13), the result is comparable to equation (3-14) and is:
The primal and dual problems for the method of geometric programming for polynomials can be stated as:
The term optimize is used for both the primal and dual problems. A polynomial can represent a cost to be minimized or a profit to be maximized, since terms of both signs are used. The results obtained from the dual problem could be a maximum, a minimum, or a saddle point since stationary points are computed. Consequently, tests from Chapter 2 or local exploration would be required to determine the character of these stationary points. Before this is discussed further let us examine the geometric programming solution or equation (3-18) to illustrate the procedure.
Obtain the geometric programming solution for equation (3-18).
The normality and orthogonality conditions are:
Solving simultaneously gives:
The optimal of y, in this case, is a maximum.
and the optimal value of x1, x2, and x3 can be computed from the definitions of the optimal weights by selecting the most convenient form from among the following:
Using the first, third and fourth, gives:
A problem can be encountered from the formulation of a geometric programming problem where the result will be negative values for the optimal weights (3,6). The dual problem required that the optimum values of y be positive. If the economic model is formulated in such a way that the optimal value is negative when calculated from the primal problem, the result will be negative weights computed in the dual problem, and it will not be possible to compute the optimum value of the function using equation (3-27). However, the value of the weights will be correct in numerical value but incorrect in sign. The previous example will be used to illustrate this difficulty, and the proof and further discussion is given by Beightler and Phillips(6).
In example 3-4 a maximum was found, and the value of the profit function was 0.1067. Had the example been to find the minimum cost, i.e., - y the result would have been -0.1067. However, this value could not have been calculated using equation (3-27). Reformulating the problem of -y with the positive terms first as:
The solution to the equation set is:
and unacceptable negative values of the optimal weights are obtained. Although not obvious, the cause of this is that the value of the function is negative at the stationary point, i.e., -0.1067. Reformulating the problem to find the stationary point of the negative of the function will give positive optimal weights and a positive value of the function as was illustrated in example 3-4.
In the illustration, example 3-4, the degree of difficulty, T - (N+1), was zero. As in posynomial optimization, the degree of difficulty must be zero or greater to be able to solve the problem by geometric programming. Also if the degree of difficulty is one or more, then the dual problem is a constrained optimization problem, which has to be solved by the procedures of Chapter 2 or other methods. However, Agognio (18) has proposed a primal-dual, normed space (PDNS) algorithm which used the primal problem and the dual problem together to locate the optimum. This algorithm consists of operations within the primal and dual programs and two sets of mappings between them which depend on a least-squares solution minimizing the two-norm of the overdetermined set of linear equations. The PDNS algorithm was tested on a number of standard problems and performed essentially equally as well as other methods. Also, the dissertation of Agogino (18) describes multiobjective optimization applications of the algorithm.
In this example we have covered the geometric programming optimization of unconstrained posynomials and polynomials. Posynomials represented the cost function of a process, and the procedure located the global minimum by solving the dual problem for the global maximum. Polynomials represented the cost or profit function of a process, and the procedure of solving the dual problem located stationary points which could be maxima, minima, or stationary points. Their character had to be determined by the methods of Chapter 2 or by local exploration. Also, for polynomials if the numerical value of the function being optimized was negative at the stationary point, this caused the optimal weights of the dual problem to be negative. It was then necessary to seek the optimum of the negative of the function to have a positive value at the stationary point. This gave positive optimal weights, and then numerical value of the function at the stationary point was computed using equation (3-27).
A complete discussion of geometric programming is given by Beightler and Phillips (6), which includes extensions to equality and inequality constraints. These extensions have the same complications as associated with the degrees of difficulty that occur with the unconstrained problems presented here. Because of these limitations the lengthy details for constraints will not be summarized here, and those who are interested in exploring this subject further are referred to the texts by Beightler and Phillips (6), and Reklaitis, et. al. (17).
The dual problem is solved when it is less complicated than the primal problem. An exponential transformation procedure for the dual problem has been described by Reklaitis et. al. (17) to make the computational problem easier when the degree of difficulty is greater than zero and also if constraints are involved. In addition, Reklaitis et. al. (17) reported on comparisons of computer codes for geometric programming optimization based on their research and that of others, including Dembo and Sarma. The testing showed that the best results were obtained with the quotient form of the generalized geometric programming problem, and second was the generalized reduced gradient solution of the exponential form of the primal problem. Also, results by Knopf, Okos, and Reklaitis (9) for batch and semicontinuous process optimization showed that the dual problem can be solved more readily than the primal problem using the generalized reduced gradient multidimensional search technique. Moreover, Phillips (10) has reported other successful applications with non-zero degrees of difficulty requiring multidimensional search methods, which are the topic of Chapter 6. In summary, if the economic model and constraints can be formulated as polynomials, there are many advantages of extensions of geometric programming which can be used for optimization.
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3-1. Solve the following problem by geometric programming
3-2.(10) Solve the following problem by geometric programming.
3-3.(13) Solve the following problem by geometric programming.
3- 4 a. Solve the following problem by geometric programming.
3-5. Solve the following problem by geometric programming.
3-6.(16) Consider the following geometric programming problem.
3-7.(11) Treatment of a waste is accomplished by chemical treatment and dilution to meet effluent code requirements. The total cost is the sum of the treatment plant, pumping power requirements, and piping cost. This cost is given by the following equation
where C is in dollars, D in inches, and Q in cfs. Find the minimum cost and best vaues of D and Q by geometric programming.
3-8. The work done by a three stage compressor is given by the following equation.
where P1 is the inlet pressure to stage 1, P2 is the discharge pressure from stage 1 and inlet pressure to stage 2, P3 is the discharge pressure from stage 2 and inlet pressure to stage 3, P4 is the discharge pressure from stage 3, and e is equal to (k-1)/k where k is the ratio of specific heats, a constant.
For specified inlet pressure P1 and volume V1 and exit pressure P4, determine intermediate pressures P2 and P3 which minimize the work by geometric programming.
The installed cost in dollars is 150 D and the lifetime pumping cost in dollars is 122,500/D5. The diameter D is in inches.
3-10. Sherwood (12) considered the optimum design of a gas transmission line, and obtained the following expression for annual charges (less fixed expenses).
where L is equal to pipe length between compressors in feet, D is the diameter in inches, F = r 0.219-1, where r is the ratio of inlet to outlet pressure. Determine the minimum cost, and the optimal values of L, F, D and r.
3-11.(15) The economic model for the annual cost is given below for a furnace in which a slag-metal reaction is to be conducted.
In this equation L is the characteristic length of the furnace in feet and T is the temperature in oK.
c. If the cost function only contained the first two terms, deleting the third term, indicate the effect on the solution by geometric programming.
3-12. The profit function for each of three chemical reactors operating in parallel with the same feed is given by the three equations below. Each reactor is operating with a different catalyst and conditions of temperature and pressure. The profit function for each reactor has the feed rates x1, x2, and x3 as the independent variable, and the parameters in the equation are determined by the catalyst and operating conditions.
3-13. A batch process has major equipment components which are a reactor, heat exchanger, centrifuge, and dryer. The total cost in dollars per batch of feed processed is given by the following equation, and it is the sum of the costs associated with each piece of equipment.
where V is the volume of feed to be processed per batch in ft3, and t1 and t2 are residence times in hours for the two sections of the process.
3-14.(6) A total of 400 cubic yards of gravel must be ferried across a river. The gravel is to be shipped in an open box of length x1, width x2, and height x3. The ends and bottom of the box cost $20/sq.yd. to build, the sides, $5/sq.yd. Runners cost $2.50/yd., and two are required to slide the box. Each round trip on the ferry cost $0.10. The problem is to find the optimal dimensions of the box that minimized the total costs of construction and transportation. This total cost is given by: