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572 Design and Optimization of Thermal Systems Using Equation (10.21) and Equation (10.24), we have w bau bau u b ab 1 2 22 (/) (/) * ** w u bau u a ab 2 2 22 * ** (/) The optimum value of the objective function is thus obtained as UF Ax w Bx w a bab b aab ** /( ) /( ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ 12 )) /( ) /( ) ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ ¤ ¦ A w B w A w bab aab 12 1 ¥¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ ww B w 12 2 (10.26) It is seen that the independent variable x is eliminated from the optimum value of the objective function. In addition, the weighting factors w 1 and w 2 are shown to indicate the relative contributions of the two terms u 1 and u 2 at the optimum. Multiple Variables The proof just given can be extended to unconstrained multiple-variable optimi- zations as long as the number of terms is greater than the number of variables by one (degree of difculty is zero) and all the terms are polynomials. The optimum of the objective function is obtained by differentiating it with respect to each of the independent variables x i , in turn, and setting the derivative equal to zero. If each of these equations is multiplied by the corresponding x i , the resulting system of equations is of the form au au au a u au nn11 1 12 2 13 3 1 1 1 21 1 0 *** , * * ! aau au a u au au nn nn 22 2 23 3 2 1 1 11 2 2 0 ** , * * ! " *** , * au a u nnnn33 1 1 0! (10.27) since there are n independent variables and n 1 terms. The coefcients a ij are the exponents, which appear as coefcients due to differentiation. By forming Geometric, Linear, and Dynamic Programming 573 a function F(x 1 , x 2 , z, x n ) as done in Equation (10.22) for a single variable and optimizing ln F, subject to w 1 w 2 z w n1 1(10.28) we get w u U u u in i ii i 3 for 1 2 1,, , (10.29) When these equations are employed with Equation (10.27), the independent variables x i are eliminated from the optimum value of the objective function U. Therefore, the optimum and the weighting factors are obtained by the geometric programming procedure outlined and applied earlier. It is seen that the weighting factors depend only on the exponents, not on the coefcients in the various terms. This means that the relative importance of each term remains unchanged as long as the exponents are the same. However, the optimum value and its location will change if the coefcients vary, for instance, because of changes in cost per unit item, energy consumption, etc. The exponents represent the dependence of the objective function on the different variables and are often xed for a given system or process. 10.1.4 CONSTRAINED OPTIMIZATION Geometric programming can also be used for optimizing systems with equality constraints. The degree of difculty is again taken as zero, so that the total num- ber of polynomial terms in the objective function and the constraints is greater than the number of independent variables by one. Let us consider the constrained optimization problem given by the objective function U u 1 u 2 u 3 (10.30) subject to the constraint u 4 u 5 1(10.31) with x 1 , x 2 , x 3 , and x 4 as the four independent variables. The unity on the right- hand side of Equation (10.31) can be obtained by normalizing the equation if a numerical term other than unity appears in the equation, which is often the case. Following the treatment given in the preceding section, the objective function and the constraint may be written as U u w u w u w ww w ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ 1 1 2 2 3 3 12 3 (10.32) 574 Design and Optimization of Thermal Systems with w 1 w 2 w 3 1and w i u U i 1 4 4 5 5 4 5 ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ u w u w w w (10.33) with w 4 w 5 1andw 4 u 4 1 , w 5 u 5 1 Equation (10.33) may be raised to the power of an arbitrary constant p, and the objective function may be written as U u w u w u w u w ww w ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ ¤ 1 1 2 2 3 3 4 4 12 3 ¦¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ pw pw u w 4 5 5 5 (10.34) Now, we may apply the method of Lagrange multipliers to obtain the opti- mum. The corresponding equations are (u 1 u 2 u 3 ) L (u 4 u 5 ) 0 u 4 u 5 1 Again, as was done in the preceding section, the derivatives are taken with respect to the independent variables x i , one at a time, and the resulting equations multiplied by x i . The constant p is arbitrary and can be taken as L/U * . Then the equations for the w’s are obtained as aw aw aw paw paw aw aw 11 1 12 2 13 3 14 4 15 5 21 1 22 0 22 233 244 255 41 1 42 2 43 0 aw paw paw aw aw aw " 33 44 4 45 5 0pa w pa w (10.35) with w 1 w 2 w 3 1andp(w 4 w 5 ) p These linear equations may be solved for w 1 , w 2 , w 3 , w 4 , w 5 , and p. Therefore, Equation (10.34) gives the optimum value of the objective function and the inde- pendent variables are obtained from the expressions for the weighting factors, as was done before. The sensitivity coefcient S c –L –pU * and has the same Geometric, Linear, and Dynamic Programming 575 physical interpretation as discussed in Chapter 8 for the Lagrange multiplier method, i.e., it is the negative of the rate of change in the optimum with respect to a change in the adjustable parameter E in the constraint G g – E 0. The preceding approach may be extended easily to more than one constraint as long as the degree of difculty is zero. The following examples illustrate the use of the method for constrained optimization. Example 10.5 For the problem considered in Example 10.4, minimize the cost of material and fabri- cation of the box for a given total volume of 5 m 3 , using geometric programming. Solution The costs of the material and fabrication vary directly as the total surface area of the rectangular container, which is open at the top. Therefore, the objective func- tion U may be taken as the area, given by U(x, y, z) xz 2xy 2yz with the constraint due to the total volume given as xyz 5 In order to apply geometric programming, the constraint is written as 0.2(xyz) 1 All the four relevant terms in the objective function and in the constraint are poly- nomials and the number of independent variables is three. Therefore, the degree of difculty D 4 – (3 1) 0. From geometric programming for constrained optimization, the optimum value of the objective function may be written as U xz w xy w yz w ww w * . ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ 12 3 12 3 22022 4 4 xyz w pw ¤ ¦ ¥ ³ µ ´ In order to eliminate the independent variables x, y, and z from the preceding equa- tion for the objective function at the optimum, we have, respectively, w 1 w 2 pw 4 0 w 2 w 3 pw 4 0 w 1 w 3 pw 4 0 Also, w 1 w 2 w 3 1 576 Design and Optimization of Thermal Systems and pw 4 p This system of linear equations may be solved easily to yield www w pw 123 1 3 44 2 3 1 and Therefore, the optimum value of the objective function is obtained as U ww w w ww w * . ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ ¤ ¦ 12202 12 3 4 12 3 ¥¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ pw 4 1 13 2 13 2 13 13 13 /// // µµ ´ ¤ ¦ ¥ ³ µ ´ 13 23 2 02 1 13 92 / / . .m The independent variables are obtained from the equations xz w U U xy w U U yz w U U 1 1 3 2 1 3 3 1 3 22 ** ** ** Therefore, these equations are solved to obtain x 2.15 m, y 1.08 m, and z 2.15 m at the optimum. Again, it can be conrmed that the area obtained at the optimum is a minimum by calculating U for small changes in x, y, and z from the opti- mum values. This simple example illustrates the use of geometric programming for constrained nonlinear optimization. Even though the requirements of polynomial expressions and zero degree of difculty limit the applicability of this approach, the method is useful in a variety of problems, particularly in thermal systems, where polynomials are frequently used to represent the characteristics. Example 10.6 In a hot-rolling process, the cost C of the system is a function of the dimensionless temperature T, the thickness ratio x, and the velocity ratio y, before and after the rolls, and is given by the expression C 1.5 5x 2 y 10 2 T subject to the constraints due to mass and energy balance given, respectively, as xy 1 and T 5x y Geometric, Linear, and Dynamic Programming 577 Formulate this optimization problem and apply geometric programming to deter- mine the optimum. Solution The constant in the objective function does not affect the optimum and the second constraint must be written in a form suitable for applying geometric programming. Therefore, the optimization problem may be written as U 5x 2 y 10 2 T subject to xy 1 and Ty x5 1 All the terms are polynomials and the degree of difculty is zero since the total number of polynomial terms is four and the number of variables is three. Therefore, the optimum value of the objective function is given by U * 510 2 1 2 23 12 13 xy wTw xy w T ww pw ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ yy xw pw 5 4 24 ¤ ¦ ¥ ³ µ ´ with the following equations for the unknowns w 1 , w 2 , w 3 , w 4 , p 1 , and p 2 : w 1 w 2 1 p 1 w 3 p 1 p 2 w 4 p 2 2w 1 p 1 w 3 p 2 w 4 0 w 1 p 1 w 3 p 2 w 4 0 2 w 2 p 2 w 4 0 where the last three equations ensure that x, y, and T, respectively, are eliminated from the expression for U * . These equations are solved to yield w 1 0.8, w 2 0.2, w 3 w 4 1, p 1 –1.2, and p 2 0.4. This gives U * ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ 5 08 10 02 1 1 1 08 02 12 55 4 976 04 ¤ ¦ ¥ ³ µ ´ . . Therefore, the optimum cost C * 1.5 4.976 6.476. Employing the equations 5x 2 y w 1 U * and 10/T 2 w 2 U * , along with the constraints, we obtain x 0.796, 578 Design and Optimization of Thermal Systems y 1.256, and T 3.170. The two Lagrange multipliers are L 1 p 1 U * 5.971 and L 2 p 2 U * 1.99, yielding corresponding sensitivity coefcients as (S c ) 1 L 1 and (S c ) 2 –L 2 . Therefore, the rst constraint is more important and an increase of 0.1 in the constant, which is unity, in the constraint will increase the dimensionless cost by 0.5971. Similarly, an increase of 0.1 in the constant in the second constraint decreases the cost by 0.199. This information can be used to adjust the design vari- ables for convenience and to use readily available items for the nal design. 10.1.5 NONZERO DEGREE OF DIFFICULTY For the application of geometric programming to the optimization of systems, we have considered only those cases where the degree of difculty D is zero. For this particular circumstance, the method requires the solution of linear equations and, consequently, provides a simple approach for optimization. However, there are obviously many problems for which the degree of difculty is not zero, as can be seen from the examples discussed in preceding chapters. If the degree of difculty is higher than zero, geometric programming can be used, but it involves solving a system of nonlinear equations. This considerably complicates the solution and it is then probably best to use some other optimization technique. Efcient computa- tional algorithms may also be developed for solving such nonlinear systems, as dis- cussed earlier in Chapter 4. Then geometric programming may be employed for a broader range of problems than if we are constrained to problems with zero degree of difculty. Inequality constraints can also be converted into equality constraints, as discussed in earlier chapters, for applying this method of optimization. Despite the possibility of solving problems with degree of difculty greater than zero, geometric programming is clearly best suited to cases where it is zero. Therefore, effort is often directed at reducing the problem with a nonzero degree of difculty to one with zero degree of difculty. One technique of achieving this is condensation, in which terms of similar characteristics may be combined to reduce the number of terms. For instance, in the rectangular container problem of Example 10.4, if an additional term 200z arises due to side supports to the box, the objective function becomes U x y z xz xy yz xyz z(,,) ( ) , 150 2 2 1 000 200 (10.36) The degree of difculty is one in this case. However, we may combine two terms, say the rst and last, to reduce the degree of difculty to zero. Writing these terms according to the geometric programming approach, we have 150 12 200 12 120 000 12 12 xz z x // (, // ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ zzxz 212 05 346 41). /. (10.37) where the two terms have been taken to be of equal importance. With this com- bined term, the degree of difculty becomes zero and the approach given in this chapter may be applied. Similarly, other terms may be combined to make the Geometric, Linear, and Dynamic Programming 579 degree of difculty zero. In some cases, information on the physical character- istics of the system may be used to eliminate relatively unimportant terms. The number of independent variables may also be reduced by holding one or more constant for the optimization in order to bring the degree of difculty to zero. All such techniques and procedures expand the application of geometric pro- gramming. For additional information on geometric programming, the references given earlier may be consulted. 10.2 LINEAR PROGRAMMING Linear programming is an important optimization technique that has been applied to a wide range of problems, particularly to those in economics, industrial engineering, power transmission, and material ow. This method is applicable if the objective functions as well as the constraints are linear functions of the independent variables. The constraints may be equalities or inequalities. Since its rst appearance about 60 years ago, linear programming has found increasing use due to the need to model, manage, and optimize large systems such as those concerned with production, trafc, and telecommunications (Hadley, 1962; Murtaugh, 1981; Dantzig, 1998; Gass, 2004; Karloff, 2006). A large number of efcient optimization algorithms for linear programming have been developed and are available commercially as well as in the public domain. For instance, MATLAB toolboxes have software that can be easily employed to solve linear programming problems for system or process optimization. The applicability of linear programming to thermal systems is somewhat lim- ited because of the generally nonlinear equations that represent these systems. However, there are problems concerned with the distribution and allocation of resources in various industries, such as manufacturing and the petroleum indus- try, which may be solved by linear programming techniques. In addition, because of the availability of efcient linear programming software, nonlinear optimiza- tion problems are solved, in certain cases, by converting these into a sequence of linear problems, as discussed in Chapter 4 for nonlinear algebraic systems. Iteration is then used, starting with an initial guessed solution, to converge to the optimum. A common method of linearization is to use the known values from the previous iteration for the nonlinear terms. For instance, an objective function of the form Uxx xx34 12 2 21 3 may be linearized as Uxx xx l l 34 12 2 21 3 () where the superscript l indicates values from the previous iteration, the others being from the current iteration. Therefore, the function becomes linear because the quantities within the parentheses are taken as known and linear program- ming can be used, with iteration, to obtain the solution. However, despite these efforts, linear programming nds its greatest use in the various areas mentioned previously, rather than in thermal engineering, for which nonlinear optimization 580 Design and Optimization of Thermal Systems techniques are often necessary. Therefore, only the essential features of this opti- mization technique and a few representative examples are given here. For further details, various references already given may be consulted. Formulation and Graphical Method The problem statement for linear programming is given in terms of the objective function and the constraints, which must both be linear functions of the inde- pendent variables. Therefore, the objective function that is to be minimized or maximized is written as Ux x x x bx bx bx bx bx nnnii (, ,, , ) 123 11 22 33 ! ii £ (10.38) subject to the constraints Gax C iijj j i £ , ,or (10.39) where b i , a ij , and C i are constants. The constraints may be equalities or inequali- ties, with G i greater or smaller than the constants C i . There are n variables and m linear equations and/or inequalities that involve these variables. In linear pro- gramming, because of inequality constraints, n may be greater than, equal to, or smaller than m, unlike the method of Lagrange multipliers, which is applicable only for equality constraints and for n larger than m. We are interested in nding the values of these variables that satisfy the given equations and inequalities and also maximize or minimize the linear objective function U. Let us illustrate the application of linear programming with the following problem involving two variables x and y: U(x, y) 5x 2y (10.40a) 4x 3y a 16 (10.40b) y 2x q4 (10.40c) This simple problem can be solved graphically, as sketched in Figure 10.4. The inequality constraints dene the feasible region in which the solution must lie. Therefore, the shaded area in the gure represents the feasible domain. The objec- tive function is dened by a family of parallel straight lines intersecting the two axes, with the value of U increasing as one moves away from the origin. Therefore, the maximum value of U is obtained by the line that touches point A, which is at the intersection of the two constraints. At this point, x 2.8, y 1.6, and U 17.2. Therefore, the optimum occurs on the boundary of the feasible domain. This is Geometric, Linear, and Dynamic Programming 581 a particular feature of linear programming and most efcient algorithms seek to move rapidly along the boundary, including the axes to obtain the optimum. Similarly, the optimum value of U may be obtained for a different set of con- straints. For instance, let us replace Equation (10.40c) by x a 2, or 4x y a 8 (10.41) In the rst case, the optimum is obtained at x 2 and y 8/3, yielding U 46/3. Again, the optimum is given by the line of constant U passing through the point given by the intersection of the two constraints. In the second case, the optimum is at x 1 and y 4, giving U 13.0. As expected, the optima occur at the bound- ary of the feasible domain. Slack Variables The preceding linear programming problems may also be solved by algebra by converting the inequalities into equalities. As mentioned in Chapter 7, additional constants, known as slack variables, may be included in order to ensure that the inequalities are not violated. Thus, by adding a constant s 1 , where s 1 0, to the left-hand side of Equation (10.40b), we can write an equation of the form 4x 3y s 1 16 (10.42) 8 y 6 x ≤ 2 A x 4x + y ≤ 8 4x + 3y ≤ 16 y – 2x ≥ –4 4 2 0 01234 FIGURE 10.4 Graphical method for solving the linear programming problems given by Equation (10.40) and Equation (10.41). [...]... velocity ratio y, and the temperature z as C 50 x y 30 0 xz 4xy 5z Using geometric programming for this unconstrained problem, obtain the minimum cost and the values of the independent variables at the optimum 10 .5 The cost C in a metal processing system is given in terms of the speed V of the material as C KS4 /3/ {V 5/ 4[2 17 .5 (3/ V)7 /3] 2} 59 4 Design and Optimization of Thermal Systems where K and S are constants... programming to find the path for minimum cost of transportation from point A to B in Figure 10.8 while passing through one of the three stations of locations C, D, and E Employ the costs given in the figure and the following costs for going between the other locations: 1–1 10 1–2 14 1 3 20 2–1 14 2–2 15 2 3 16 3 1 20 3 2 16 3 3 15 590 Design and Optimization of Thermal Systems Solution Starting with point A,... C1, C2, or C3, resulting in three different subsections Similarly, D2 and D3 can be reached by going through these three stations of C Therefore, the total cost in going from A to D is given by Through C1 C2 C3 Cost to D1 27 29 38 Cost to D2 31 30 34 Cost to D3 37 31 33 The optimal path for each subsection is underlined We can similarly consider reaching E1, E2, and E3 through D1, D2, and D3 In each... possible mainly because of the availability of efficient computational schemes for analysis and fast computers These ideas have been extended to the optimization of topology, profile, trajectory, and configuration in different types of systems and applications Though an active area for research in the design of structures, shape optimization has not been used very much in thermal systems and processes Several... illustrates the use of the simplex algorithm Example 10.8 The objective function for an optimization problem is taken as the total income, which involves an income of five units on item A and seven units on item B The former Geometric, Linear, and Dynamic Programming 58 5 requires 2 .5 hours of cutting and 1 .5 hours of polishing, whereas item B requires 4 hours of cutting and 1 hour of polishing If the... Optimization of Thermal Systems subjected to the constraints x1 3x2 35 x2 18 x1 5. 5x1 2.5x2 110 x1 0 x2 0 10.17 The number of components produced by a company in two different categories are x and y The objective function is the overall income U, given by U 1.25x 1.75y x y 450 2x 6y 750 4x 7y 1480 subjected to the constraints Solve this problem by any linear programming method to obtain the number of components... this optimization technique to thermal processes and systems is rather limited because it is often difficult to divide continuous processes into steps When stages do arise, as in heating and cooling systems, the number of stages is often small and is generally not interchangeable or movable within the process However, dynamic programming is useful in the economic analysis and management of thermal systems. .. of 58 4 Design and Optimization of Thermal Systems y 150 3x + 2y ≤ 30 0 100 50 x + 3y ≤ 200 x 0 50 100 150 200 FIGURE 10.6 Graphical solution to the linear programming problem posed in Example 10.7 the pivot element to obtain unity, as well as eliminating the other coefficients in the column containing the pivot element, are used during the solution procedure The method uses a tabular form of data presentation... seen that the cheapest is through E2 and has a total cost of 55 Therefore, the optimal path is given as A – C1 – D1 – E2 – B By using the optimal solutions obtained earlier for the subsections of the overall path, the number of computations is reduced The total number of combinations is 3 3 3 1 27 The number of calculations needed here is 9 9 3 21 Clearly, the benefit of dynamic programming in reducing... extraction of the optimum by linear programming Example 10.7 A company produces x quantity of one product and y of another, with the profit on the two items being four and three units, respectively Item 1 requires one hour of facility A and three hours of facility B for fabrication, whereas item 2 requires three hours of facility A and two hours of facility B, as shown schematically in Figure 10 .5 The total . 12 2 13 3 14 4 15 5 21 1 22 0 22 233 244 255 41 1 42 2 43 0 aw paw paw aw aw aw " 33 44 4 45 5 0pa w pa w (10 . 35 ) with w 1 w 2 w 3 1andp(w 4 w 5 ) p These. Cost to D2 Cost to D3 C1 27 31 37 C22 930 31 C 338 3 433 The optimal path for each subsection is underlined. We can similarly consider reaching E1, E2, and E3 through D1, D2, and D3. In each case,. function and the constraint may be written as U u w u w u w ww w ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ 1 1 2 2 3 3 12 3 (10 .32 ) 57 4 Design and Optimization of Thermal Systems with w 1 w 2 w 3
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