Reformulation of the modified goal programming for logarithmic piecewise linear function

Abstract

The purpose of this paper is to present a reformulation of the model presented by Chang [A modified goal programming model for piecewise linear function, European Journal of Operational Research 139 (2002) 62–67]. The modified goal programming (MGP) model of Chang has been accepted as an efficient method published for solving polynomial/posynomial (P/P) programming problems. However, it still cannot be used to fully practical applications because the problem with negative variable is undefined in logarithmic piecewise linear function (LPLF). This paper presents some appropriate strategies to overcome this undefined problem. The reformulation MGP model can be used to solve P/P programming problems with negative/positive variables. In addition, an illustrative example is included for demonstrating the correctness of the proposed method.

Introduction

The goal programming (GP) stems from the work of Charnes and Cooper [8], with further development by Lee [9], Ignizio [10], among others. Tamiz et al. [11] provided a comprehensive review of GP. The overall purpose of GP is to minimize the deviations between the achievement of goals and their aspiration levels. A typical linear GP is expressed as follows:

• (P1)

$$\begin{array}{cc}\hfill & \mathrm{Minimize}\sum _{i=1}^n(d_i^{+}+d_i^{-})\hfill \\ \hfill & \mathrm{Subject}\mathrm{to}\hfill \\ \hfill & f_i(X)-d_i^{+}+d_i^{-}=g_i\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}\hfill \\ \hfill & d_i^{+}\text{,}{d}_i^{-}⩾0\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}\hfill \\ \hfill & X\in F(F\mathrm{is}\mathrm{a}\mathrm{feasible}\mathrm{set})\text{.}\hfill \end{array}$$

where f(X) is the linear function of the ith goal; g_i is the aspiration level of the ith goal; $d_i^{+}$ and $d_i^{-}$ are, respectively, over- and under-achievement of the ith goal.

For reducing the additional continuous variables $d_i^{+}$ and $d_i^{-}$, Li [12] proposed an equivalent model for solving linear GP as follows.

• (P2)

$$\begin{array}{cc}\hfill & \mathrm{Minimize}\sum _{i=1}^n(2d_i-f_i(X)+g_i)\hfill \\ \hfill & \mathrm{Subject}\mathrm{to}\hfill \\ \hfill & -f_i(X)+d_i+g_i⩾0\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}\hfill \\ \hfill & d_i⩾0\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}\hfill \\ \hfill & X\in F(F\mathrm{is}\mathrm{a}\mathrm{feasible}\mathrm{set})\text{,}\hfill \end{array}$$

where the positive deviation of the ith goal is d_i, and the negative deviation of ith goal is −f_i(X) + d_i + g_i.

Logarithmic piecewise linear function (LPLF) is an important technique for solving non-linear programming problems. This method has been applied to many applications including posynomial fractional programming, fuzzy multiobjective, assortment, and regression analysis problems [2], [3], [5], [7]. The LPLF can be stated as follows. Let f(x_i) be a LPLF of x_i, where a_i, i = 1, 2, …, n, are the break points of f(x_i), and s_i, i = 1, 2, …, n, are the slopes of line segments between a_i and a_i+1. f(x_i) can be expressed as the following sum of absolute terms.

$$f(x_i)=a_0+s_1(x_i-a_1)+\sum _{i=2}^{n-1}\frac{s_i-s_{i-1}}{2}(|x_i-a_i|+x_i-a_i)\text{,}$$

f(x_i) can be easily applied to any P/P programming problem. Chang [1] proposed an efficient technique called modified GP (MGP) model for solving f(x_i), requiring least number of auxiliary constraints and additional continuous variables as follows.

• (P3)

$$\begin{array}{cc}\hfill & \mathrm{Minimize}\sum _{i=1}^{n}2\left(x_i-a_i+\sum _{k=1}^j{d}_k\right)\hfill \\ \hfill & \mathrm{Subject}\mathrm{to}\hfill \\ \hfill & x_i+\sum _{i=1}^m{d}_k⩾a_n\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}\hfill \\ \hfill & a_{i-1}⩽d_i⩽a_i\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}\hfill \\ \hfill & X\in F(F\mathrm{is}\mathrm{a}\mathrm{feasible}\mathrm{set})\text{.}\hfill \end{array}$$

For instance consider the following LPLF problem, which can be formulated by various approaches.

Example 1

$$\begin{array}{cc}\hfill & \mathrm{Minimize}(|x_1-1|+x_1-1)+(|x_1-2|+x_1-2)\hfill \\ \hfill & \mathrm{Subject}\mathrm{to}\hfill \\ \hfill & X\in F(F\mathrm{is}\mathrm{a}\mathrm{feasible}\mathrm{set})\text{.}\hfill \end{array}$$

Example 1 can be formulated as the following three equivalent programs using Charnes and Cooper, Li, and Chang’s methods.
(Charnes and Cooper’s method)

$$\begin{array}{cc}\hfill & \mathrm{Minimize}{d}_1^{+}+d_1^{-}+x_1-1+d_2^{+}+d_2^{-}+x_1-2\hfill \\ \hfill & \mathrm{Subject}\mathrm{to}\hfill \\ \hfill & x_1-d_1^{+}+d_1^{-}-1=0\text{,}\hfill \\ \hfill & x_1-d_2^{+}+d_2^{-}-2=0\text{,}\hfill \\ \hfill & X\in F(F\mathrm{is}\mathrm{a}\mathrm{feasible}\mathrm{set})\text{,}\hfill \end{array}$$

where $d_1^{+}⩾0\text{,}{d}_1^{-}⩾0\text{,}{d}_2^{+}⩾0$, and $d_2^{-}⩾0$.
(Li’s method)

$$\begin{array}{cc}\hfill & \mathrm{Minimize}2d_1+2d_2\hfill \\ \hfill & \mathrm{Subject}\mathrm{to}\hfill \\ \hfill & -x_1+d_1+1⩾0\text{,}\hfill \\ \hfill & x_2-d_2^{+}+d_2^{-}-2=0\text{,}\hfill \\ \hfill & X\in F(F\mathrm{is}\mathrm{a}\mathrm{feasible}\mathrm{set})\text{,}\hfill \end{array}$$

where $d_1⩾0$ and d₂ ⩾ 0.
(Chang’s method)

$$\begin{array}{cc}\hfill & \mathrm{Minimize}2(x_1-1+d_1)+2(x_1-2+d_1+d_2)\hfill \\ \hfill & \mathrm{Subject}\mathrm{to}\hfill \\ \hfill & x_1+d_1+d_2⩾2\text{,}\hfill \\ \hfill & X\in F(F\mathrm{is}\mathrm{a}\mathrm{feasible}\mathrm{set})\text{,}\hfill \end{array}$$

where 0 ⩽ d₁ ⩽ 1 and 0 ⩽ d₂ ⩽ 1.

From the above we realize that Chang’s method is the most efficient approach for solving LPLF problem, requiring least number of auxiliary constraints. However, the MGP model of Chang still cannot be used to solve P/P problem with negative variables because of negative variable is undefined in the LPLF. The purpose of this paper is to present a reformulation of the MGP (RMGP) model presented by Chang [1]. The RMGP model can be easily applied to P/P programming problems with negative and/or positive variables.