Hybrid particle swarm optimization and differential evolution algorithm for bi-level programming problem and its application to pricing and lot-sizing decisions

Abstract

This paper proposes a hierarchical hybrid particle swarm optimization (PSO) and differential evolution (DE) based algorithm (HHPSODE) to deal with bi-level programming problem (BLPP). To overcome the shortcomings of basic PSO and basic DE, this paper improves PSO and DE, respectively by using a velocity and position modulation method in PSO and a modified mutation strategy in DE. HHPSODE employs the modified PSO as a main program and the modified DE as a subprogram. According to the interactive iterations of modified PSO and DE, HHPSODE is independent of some restrictive conditions of BLPP. The results based on eight typical bi-level problems demonstrate that the proposed algorithm HHPSODE exhibits a better performance than other algorithms. HHPSODE is then adopted to solve a bi-level pricing and lot-sizing model proposed in this paper, and the data is used to analyze the features of the proposed bi-level model. Further tests based on the proposed bi-level model also exhibit good performance of HHPSODE in dealing with BLPP.

Appendix

Test problems

T1	T 2
$\begin{array}{l} \max \;f_1 =-2x_1 +11x_2 \;\text{ where }\;x_2 \;\text{ solves } \\ \text{ max }\;f_2 =-x_1 -3x_2 \\ \text{ s.t. } x_1 -2x_2 \le 4 \\ \quad \,2x_1 -x_2 \le 24 \\ \quad \,3x_1 +4x_2 \le 96 \\ \quad \,x_1 +7x_2 \le 126 \\ \quad \,-4x_1 +5x_2 \le 65, \\ \quad \,x_1 +4x_2 \ge 8 \\ \quad \,x_1 \ge 0,x_2 \ge 0 \\ \end{array}$	$\begin{array}{l} \max \;f_1 =x_2 \;\text{ where }\;x_2 \;\text{ solves } \\ \text{ max }\;f_2 =x_2 \\ \text{ s.t. } -x_1 -2x_2 \le 10 \\ \quad \,x_1 -2x_2 \le 6 \\ \quad \,2x_1 +x_2 \le 21 \\ \quad \,x_1 +2x_2 \le 38, \\ \quad \,-x_1 +2x_2 \le 18, \\ \quad \,x_1 \ge 0,x_2 \ge 0 \\ \end{array}$
T3	T4
$\begin{array}{l} \max \;f_1 =3x_2 \;\text{ where }\;x_2 \;\text{ solves } \\ \text{ max }\;f_2 =-x_2 \\ \text{ s.t. } -x_1 +x_2 \le 3 \\ \quad \,x_1 -2x_2 \le 12 \\ \quad \,4x_1 +x_2 \le 12 \\ \quad \,x_1 \ge 0,x_2 \ge 0 \\ \end{array}$	$\begin{array}{l} \max \;f_1 =8x_1 +4x_2 -4y_1 +40y_2 +4y_3 \\ \text{ where }\;y\;\text{ solves } \\ \text{ max }\;f_2 =-x_1 -2x_2 -y_1 -y_2 -2y_3 \\ \text{ s.t. } y_1 -y_2 -y_3 \ge -1 \\ \quad \,-2x_1 +y_1 -2y_2 +0.5y_3 \ge -1 \\ \quad \,-2x_1 -2y_1 +y_2 +0.5y_3 \ge -1 \\ \quad \,x_1 ,x_2 ,y_1 ,y_2 ,y_3 \ge 0 \\ \end{array}$
T5	T6
$\begin{array}{l} \min \;f_1 =-x_1^2 -3x_2 -4y_1 y_2^2 \\ \text{ s.t. } x_1^2 +2x_2 \le 4,x_1 \ge 0,x_2 \ge 0 \\ \text{ where }\;y\;\text{ solves } \\ \text{ min }\;f_2 =2x_1^2 +y_1^2 -5y_2 \\ \text{ s.t. } x_1^2 -2x_1 +x_2^2 -2y_1 +y_2 \ge -3 \\ \quad \, x_2 +3y_1 -4y_2 \ge -4,y_1 \ge 0,y_2 \ge 0 \\ \end{array}$	$\begin{array}{l} \min \;f_1 =x^{2}+(y-10)^{2} \\ \text{ s.t. } x+2y-6\le 0,-x\le 0 \\ \text{ where }\;y\;\text{ solves } \\ \text{ min }\;f_2 =x^{3}+2y^{3}+x-2y-x^{2} \\ \text{ s.t. } -x-2y-3\le 0,-y\le 0 \\ \end{array}$
T7	T8
$\begin{array}{l} \min \;f_1 =(x-5)^{4}+(2y+1)^{4} \\ \text{ s.t. } x+y-4\le 0,-x\le 0 \\ \text{ where }\;y\;\text{ solves } \\ \text{ min }\;f_2 =e^{-x+y}+x^{2}+2xy+y^{2}+2x+6y \\ \text{ s.t. } -x+y-2\le 0,-y\le 0 \\ \end{array}$	$\begin{array}{l} \min \;f_1 =(x_1 -y_2 )^{4}+(y_1 -1)^{2}+(y_1 -y_2 )^{2} \\ \text{ s.t. } -x_1 \le 0 \\ \text{ where }\;y\;\text{ solves } \\ \text{ min }\;f_2 =2x_1 +e^{y_1 }+y_1^2 +4y_1 +2y_2^2 -6y_2 \\ \text{ s.t.6 }x_1 +2y_1^2 +e^{y_2 }-15\le 0,-y_1 \le 0,y_1 -4\le 0 \\ \quad \,5x_1 +y_1^4 +y_2 -25\le 0,-y_2 \le 0,y_2 -2\le 0 \\ \end{array}$