Posynomial geometric programming with parametric uncertainty

European Journal of Operational Research 168 (2006) 345–353 www.elsevier.com/locate/ejor

Continuous Optimization

Posynomial geometric programming with parametric uncertainty Shiang-Tai Liu

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Graduate School of Business and Management, Vanung University, Chung-Li, Tao-Yuan 320, Taiwan, ROC Received 28 May 2003; accepted 29 April 2004 Available online 3 August 2004

Abstract Geometric programming provides a powerful tool for solving nonlinear problems where nonlinear relations can be well presented by exponential or power function. This paper develops a procedure to derive the lower and upper bounds of the objective of the posynomial geometric programming problem when the cost and constraint parameters are uncertain. The imprecise parameters are represented by ranges, instead of single values. An imprecise geometric program is transformed to a family of conventional geometric programs to calculate the objective value. The derived result is also in a range, where the objective value would appear. The ability of calculating the bounds of the objective value developed in this paper might help lead to more realistic modeling efforts in engineering design areas.
2004 Elsevier B.V. All rights reserved. Keywords: Geometric programming; Uncertain parameter; Duality theorem; Optimization

1. Introduction Geometric programming is a methodology for solving algebraic nonlinear optimization problems. Its attractive structural properties as well as its elegant theoretical basis have led to a number of interesting applications and the development of numerous useful results. The integrated circuit design [5,11,18], engineering design [4,21], project management [26], and inventory management [3,12,13,17,25,27] are the examples. One of the remarkable properties of geometric programming is that a problem with highly nonlinear constraints can be stated equivalently as one with only linear constraints. This is because there is a strong

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2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.04.046

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S.-T. Liu / European Journal of Operational Research 168 (2006) 345–353

duality theorem for geometric programming problems. If the primal problem is in posynomial form, then a global minimizing solution to that problem can be obtained by solving the dual maximization. The dual constraints are linear, and linearly constrained programs are generally easier to solve than ones with nonlinear constraints. This allows for the development of powerful solution techniques for geometric programs. Efficient and effective algorithms [1,2,8–10,14,15,19,20,22–24,28,29] have been developed for solving the geometric programming problems when the cost and constraint coefficients are known. As in postoptimality analysis in linear programming, [6,7,16] address the issue of sensitivity analysis in geometric programming. Nevertheless, the parameters in the problem are handled indirectly and are not allowed to vary simultaneously while calculating the bounds of the objective value. Many applications of geometric programming are engineering design problems in which some of the problem parameters are estimates of actual values [2]. There are also cases in which these coefficients may not be presented in a precise manner. For example, in project management the time required to complete the various activities in a research and development project may be only known approximately. In determining the inventory policy of a novel technology product, the demand and supply quantities may be uncertain due to insufficient market information and are specified by ranges. If some parameters are imprecise or uncertain, then the most likely values are usually adopted to make the conventional geometric program workable. This simplification might result in a derived result which is misleading. One way to manipulate imprecise parameters is via probability distributions. However, a probability distribution requires construction of a prior predictable regularity or a posterior frequency determination, which may not be possible in certain cases. An alternative is to apply interval estimates to represent the uncertain parameters, instead of single values. Clearly, when the parameters in the problem are imprecise, the objective value will be imprecise as well. In this paper, we restrict attention to posynomial geometric programs and develop a solution procedure that is able to calculate the optimal objective value for the problem, where at least one of the parameters is an imprecise number. A pair of two-level mathematical programs is formulated to calculate the lower and upper bounds of the objective value. The rest of this paper is organized as follow. First we first briefly state the posynomial geometric programming problem with imprecise parameters. Then a pair of two-level mathematical programs for calculating the bounds of the objective value is formulated; we use two examples to explain the idea of this paper. Finally, some conclusions are drawn from the discussion.

2. Imprecise geometric programming A constrained posynomial geometric program is an optimization problem of the following form: Min x

s:t:

f 0 ðxÞ ¼

s0 X

c0t

t¼1

f i ðxÞ ¼

si X t¼1

xj > 0;

n Y

c

xj 0tj

j¼1

cit

n Y

c

xj itj 6 1;

i ¼ 1; . . . ; m;

ð1Þ

j¼1

j ¼ 1; . . . ; n:

The posynomial f0(x) containing s0 terms is the objective function, while the posynomials fi(x) for i = 1, 2, . . ., m containing si terms represent m inequality constraints. By the definition of posynomial, all the coefficients cit for i = 0, 1, . . ., m and t = 1, . . ., sm are positive. In this model all the coefficients must be precise. Intuitively, if any of the coefficients is imprecise, the objective value should be imprecise as well.

S.-T. Liu / European Journal of Operational Research 168 (2006) 345–353

347

That is, the objective value should also appear in a range. Furthermore, when the right-hand side value becomes an interval number rather than a single value, it is not so straightforward to convert the constraint to the standard form like Model (1). Suppose we modify the right-hand sides of the constraints in the geometric program (1) as follows: Min x

s:t:

s0 X

c0t

n Y

t¼1

c

xj 0tj

j¼1

si X

n Y

cit

t¼1

c

xj itj 6 bi ;

i ¼ 1; . . . ; m;

ð2Þ

j¼1

j ¼ 1; . . . ; n;

xj > 0;

where all bi are positive numbers. If bi = 1, " i, then this modified geometric program coincides with the original one. Otherwise, the constraints need some amendment to be consistent with Model (1). Let ^c0t , ^cit , and ^ bi denote the interval counterparts of c0t, cit, and bi, respectively. The posynomial geometric programming problem with imprecise parameters is of the following form: Min x

s:t:

s0 X

^c0t

t¼1 si X

n Y

c

xj 0tj

j¼1

^cit

t¼1

n Y

c

xj itj 6 ^ bi ;

i ¼ 1; . . . ; m;

ð3Þ

j¼1

xj > 0;

j ¼ 1; . . . ; n;

L L L U L ^ cit 2 ½C Lit ; C U where ^c0t 2 ½C L0t ; C U 0t , ^ it , bi 2 ½Bi ; Bi , C 0t > 0, C it > 0, and Bi > 0, " i,t. Suppose we are interested in deriving the bounds of the objective value of Model (3). The major difficulty lies on how to deal with the varying ranges of the parameters in the objective function and the constraints. Let L U ^ S ¼ fð^c; ^ bÞjC Lit 6 ^cit 6 C U it ; Bk 6 bk 6 Bk ; 1 6 t 6 si ; i ¼ 0; 1; . . . ; m; k ¼ 1; 2; . . . ; mg:

For each ð^c; ^ bÞ 2 S, we denote Zð^c; ^ bÞ to be the objective value of Model (3). Let ZL and ZU be the min^ imum and the maximum of Zð^c; bÞ on S, respectively, namely Z L ¼ MinfZð^c; ^ bÞjð^c; ^ bÞ 2 Sg; U ^ c; bÞ ^ 2 Sg; Z ¼ MaxfZð^c; bÞjð^ which can be reformulated as the following pair of two-level mathematical programs: Z L ¼ Min Min ð^c;^ bÞ2S

x

s:t:

s0 X

n Y

^c0t

t¼1

j¼1

si X

n Y

^cit

t¼1

xj > 0;

c

xj 0tj c xj itj 6 ^ bi ;

i ¼ 1; . . . ; m;

j¼1

j ¼ 1; . . . ; n;

ð4aÞ

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S.-T. Liu / European Journal of Operational Research 168 (2006) 345–353

Z U ¼ Max Min ð^c;^ bÞ2S

s0 X

x

s:t:

n Y

^c0t

t¼1

c

xj 0tj

j¼1

si X

^cit

n Y

t¼1

c xj itj 6 ^ bi ;

i ¼ 1; . . . ; m;

ð4bÞ

j¼1

j ¼ 1; . . . ; n:

xj > 0;

In Model (4) the right-hand side value ^ bi may not be equal to the constant value 1. In this case, one can divide the constraint coefficients ^cit by the right-hand side value ^bi , " i, to be the following standard geometric program form: Z L ¼ Min Min ð^c;^ bÞ2S

s0 X

x

s:t:

^c0t

t¼1

n Y

c

xj 0tj

j¼1

si X

1 ^cit ð^ bi Þ

n Y

t¼1

ð^c;^ bÞ2S

s0 X

x

s:t:

i ¼ 1; . . . ; m;

ð5aÞ

j¼1

j ¼ 1; . . . ; n;

xj > 0; Z U ¼ Max Min

c

xj itj 6 1;

^c0t

t¼1

n Y

c

xj 0tj

j¼1

si X

1 ^cit ð^ bi Þ

t¼1

n Y

c

xj itj 6 1;

i ¼ 1; . . . ; m;

ð5bÞ

j¼1

xj > 0;

j ¼ 1; . . . ; n:

Model (5a) is to find the minimum value against the best possible value on S. To derive the lower bound of the objective value in Model (5a), one can directly set all ^c0t to their lower bounds C L0t in the objective function. Moreover, since the values of the right-hand sides are the constant 1, the lower the ratios of ^cit =^ bi in constraints, the larger the feasible region is. Therefore, the values of ^cit and ^bi should, respectively, set to its lower bound C Lit and its upper bound BU i , " i, t. Hence, we can transform Model (5a) to the following mathematical form: Z L ¼ Min x

s:t:

s0 X

C L0t

t¼1 si X

n Y

c

xj 0tj

j¼1

1 C Lit ðBU i Þ

t¼1

xj > 0;

n Y

c

xj itj 6 1;

i ¼ 1; . . . ; m;

ð6Þ

j¼1

j ¼ 1; . . . ; n:

Model (6) is a conventional geometric programming problem. In the literature the solution techniques for geometric program may be categorized as either primal-based algorithms that directly solve the nonlinear primal problems, or dual-based algorithms that solve the equivalent linearly constrained dual [20]. In view of Rajgopal and Bricker [24], the dual problem has the desirable features of being linearly constrained and having an objective function with attractive structural properties, thus making it a natural candidate solution. According to Beightler and Phillips [2] and Duffin et al. [9], one can transform Model (6) to the corresponding dual geometric program as follows:

S.-T. Liu / European Journal of Operational Research 168 (2006) 345–353

Z L ¼ Max w

s:t:

349

s0 si m Y Y Y w

1 wit ðC L0t =w0t Þ 0t ðC Lit ðBU i Þ wi0 =wit Þ t¼1 s0 X

i¼1 t¼1

w0t ¼ 1;

ð7Þ

t¼1 si m X X i¼0

citj wit ¼ 0;

j ¼ 1; . . . ; n;

t¼1

8i; t;

wit P 0;

P si wit ¼ wi0 . Model (7) is to find a stationary point of Lagrangian function for a concave objective where t¼1 function subject to a set of convex constraints. Hence, Model (7) has a unique stationary point of Lagrangian function––a global maximum [2,9]. Thus we can derive the lower bound of the objective value by solving Model (7). Model (5b) is to find the maximum value among the best possible objective values over all decision variables. To solve Model (5b), the dual of the inner level problem is formulated to become a maximization problem to be consistent with the maximization operation of outer level. If the dual problem is feasible, the it is well-known from the duality theorem of geometric programming that the primal model and the dual model have the same optimal objective value. Thus Model (5b) becomes: Z U ¼ Max Max ð^c;^ bÞ2S

w

s:t:

s0 si m Y Y Y ð^c0t =w0t Þw0t ð^cit ð^ bi Þ 1 wi0 =wit Þwit t¼1 s0 X

i¼1 t¼1

w0t ¼ 1;

ð8Þ

t¼1 m X

si X

i¼0

t¼1

citj wit ¼ 0;

wit P 0;

j ¼ 1; . . . ; n;

8i; t:

Since both inner level and outer level perform the same maximization operation and variables ^c0t , ^cit and U ^ bi are all in the objective function, one can set all ^c0t and ^cit to their upper bounds C U 0t and C it , respectively. L ^ On the other hand, we should set all bi to their lower bounds Bi . Consequently, the two-level mathematical program in (8) can be simplified to the following conventional dual geometric program: Z U ¼ Max w

s:t:

s0 si m Y Y Y w0t wit L 1 ðC U =w Þ ðC U 0t 0t it ðBi Þ wi0 =wit Þ t¼1 s0 X

i¼1 t¼1

w0t ¼ 1;

t¼1 m X

si X

i¼0

t¼1

wit P 0;

citj wit ¼ 0;

ð9Þ j ¼ 1; . . . ; n;

8i; t:

The lower bound ZL and upper bound ZU of the objective value are directly solved from Models (7) and (9), respectively. In this paper, the parameters in the problem are allowed to vary simultaneously. When all the parameters are crisp, Models (7) and (9) become identical and can be reduced to the dual form of Model (1), a conventional geometric program.

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3. Examples In this section, we use two geometric programming problems to illustrate the methodology proposed in this paper. Example 1. The geometric programming problem considered by Dinkle and Tretter [7] is to perform sensitivity analysis using interval arithmetic. The associated mathematical form is

1=2 1 t3

Min

ð20; 70Þt 1 1 t2

s:t:

1 2 2 4 1=2 1 t t þ t2 t3 6 1; 3 1 2 3

t

þ 20t1 t3 þ 20t1 t2 t3 ð10Þ

t1 ; t2 ; t3 > 0; where (20, 70) represents the interval of the coefficient in the objective function. According to Models (7) and (9), the problem can be transformed to the following pair of geometric programs:  w01  w02  w03  w  w 20 20 20 w11 þ w12 11 4ðw11 þ w12 Þ 12 Z L ¼ Max w w01 w02 w03 3w12 3w11 s:t:

w01 þ w02 þ w03 ¼ 1;

w01 þ w02 þ w03 2w11 ¼ 0; 1 1

w01 þ w03 2w11 þ w12 ¼ 0; 2 2

w01 þ w02 þ w03 w12 ¼ 0;

ð11aÞ

w01 ; w02 ; w03 ; w11 ; w12 P 0; Z U ¼ Max w

s:t:



70 w01

w01 

20 w02

w02 

20 w03

w03  w  w w11 þ w12 11 4ðw11 þ w12 Þ 12 3w12 3w11

w01 þ w02 þ w03 ¼ 1;

w01 þ w02 þ w03 2w11 ¼ 0; 1 1

w01 þ w03 2w11 þ w12 ¼ 0; 2 2

w01 þ w02 þ w03 w12 ¼ 0;

ð11bÞ

w01 ; w02 ; w03 ; w11 ; w12 P 0: Using the logarithmic form of the objective function, both Problems (11a) and (11b) have concave objective functions with linear constraints. Hence, one can derive the global optimum solution by solving Problem (11). The lower bound of the objective value ZL = 90 occurs at w 01 ¼ 0:1111, w 02 ¼ 0:4444, w 03 ¼ 0:4445, w 11 ¼ 0:3889, and w 12 ¼ 0:7778. In the constrained geometric programming problem, the dual optimal solutions w* provide weights of the terms in the constraints of transformed primal problem. After transformation of w* [2], one derives the corresponding primal solution t 1 ¼ 1, t 2 ¼ 1, and t 3 ¼ 2. The upper bound of the objective value ZU = 115 occurs at w 01 ¼ 0:3044, w 02 ¼ 0:3478, w 03 ¼ 0:3478, w 11 ¼ 0:1957, w 12 ¼ 0:3913, and one can obtain the associated primal solution t 1 ¼ 1, t 2 ¼ 1, and t 3 ¼ 2. The derived optimal solutions for lower and upper bounds of the objective value are the same as those reported in Dinkle and Tretter [7]. Nevertheless, the method of Dinkle and Tretter cannot allow the interval

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351

parameters (cost or constraint coefficients) in the problem to be changed simultaneously. In the next example, we shall let all the imprecise parameters in the problem vary simultaneously to calculate the bounds of the objective value. Example 2. Consider the geometric programming problem with imprecise parameters in the objective function and constraints as follows: Min

1

3 2 2 ð2; 3Þx21 x 1 2 x3 x4 þ ð4; 4:2Þx1 x2 x3

s:t:

1 ð3; 3:6Þx31 x3 þ x 1 1 x3 6 ð2; 4Þ;

x

2 x 1 2 x3 x4 þ ð2; 2:8Þx1 x2 x4 6 1; x1 ; x2 ; x3 > 0:

ð12Þ

To find the lower and upper bounds of the objective value, the problem is transformed into the following pair of geometric programs to be solved:  w01  w02  w  w  w  w 2 4 3w10 11 w10 12 w20 21 2w20 22 L Z ¼ Max w w01 w02 4w11 4w12 w21 w22 s:t: w01 þ w02 ¼ 1; 2w01 3w02 þ 3w11 w12 þ 2w22 ¼ 0;

w01 þ 2w02 w21 þ w22 ¼ 0;

ð13aÞ

w01 2w02 þ w11 w12 þ w21 ¼ 0;

w01 þ w21 þ w22 ¼ 0; w01 ; w02 ; w11 ; w12 ; w21 ; w22 P 0; where w10 = w11 + w12, w20 = w21 + w22;  w01  w02  w  w  w  w 3 4:2 3:6w10 11 w10 12 w20 21 2:8w20 22 U Z ¼ Max w w01 w02 2w11 2w12 w21 w22 s:t: w01 þ w02 ¼ 1; 2w01 3w02 þ 3w11 w12 þ 2w22 ¼ 0;

w01 þ 2w02 w21 þ w22 ¼ 0;

ð13bÞ

w01 2w02 þ w11 w12 þ w21 ¼ 0;

w01 þ w21 þ w22 ¼ 0; w01 ; w02 ; w11 ; w12 ; w21 ; w22 P 0; where w10 = w11 + w12, w20 = w21 + w22. The lower bound of the objective value ZL = 6.8573 occurs at x 1 ¼ 0:8853, x 2 ¼ 0:2484, x 3 ¼ 0:3440,

x4 ¼ 0:5636, and the upper bound of the objective value ZU = 10.8301 occurs at x 1 ¼ 0:5078, x 2 ¼ 0:6123, x 3 ¼ 1:5536, x 4 ¼ 0:3356. Clearly, if the parameters vary in ranges, then the objective value derived is also in a range.

4. Conclusions Since its emergence in the 1960s, geometric programming has undergone considerable development, has experienced a proliferation of proposals for numerical solution algorithms, and has led to

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S.-T. Liu / European Journal of Operational Research 168 (2006) 345–353

considerable practical engineering applications. Most of the studies are concerned with deterministic cases; i.e., all parameters can be assigned precisely. This paper develops a method to find the objective value when some parameters in the problem are imprecise numbers. The idea is to transform an imprecise geometric program to a family of conventional geometric programs, which are utilized to calculate the lower and upper bounds of the objective value. As a matter of fact, an interval estimation is more informative than a point estimation of the objective value for cases of imprecise parameters. Two examples are employed for illustration in this study. The derived results show that the method proposed in this paper has generality in handling geometric programs with imprecise parameters. Since both two examples have only one degree of difficulty, the optimal solution can be easily discovered. However, as the degree of difficulty increases, solution becomes harder, requiring more efficient solution techniques. In this case, we can refer the reader to the studies [2,15,24] that comprehensively discuss algorithms and computational aspects for geometric programming problems. Geometric programming has already shown its power in practice in the past. In real-world applications, the parameters in the geometric program may not be known precisely due to insufficient information. With the ability of calculating the bounds of the objective value developed in this paper, it might help spawn wider applications.

Acknowledgements This research is supported by the National Science Council of Republic of China under Contract NSC92-2416-H-238-004. The author would like to thank Dr. Yung-Yih Lur and the referees for their helpful comments and suggestions.

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