Evolutionary Bi-objective Optimization for Bulldozer and Its Blade in Soil Cutting

Abstract

An evolutionary optimization approach is adopted in this paper for simultaneously achieving the economic and productive soil cutting. The economic aspect is defined by minimizing the power requirement from the bulldozer, and the soil cutting is made productive by minimizing the time of soil cutting. For determining the power requirement, two force models are adopted from the literature to quantify the cutting force on the blade. Three domain-specific constraints are also proposed, which are limiting the power from the bulldozer, limiting the maximum force on the bulldozer blade and achieving the desired production rate. The bi-objective optimization problem is solved using five benchmark multi-objective evolutionary algorithms and one classical optimization technique using the ε-constraint method. The Pareto-optimal solutions are obtained with the knee-region. Further, the post-optimal analysis is performed on the obtained solutions to decipher relationships among the objectives and decision variables. Such relationships are later used for making guidelines for selecting the optimal set of input parameters. The obtained results are then compared with the experiment results from the literature that show a close agreement among them.

Appendix

Model of [6] for Cutting Force on Wide Blade. Nomenclature and description of equations can be found in [22]

$${\text{m}}_{1} {\text{g}} = \frac{1}{2}{{\upgamma }}_{\text{o }} {\text{B}}\left( {{\text{H }} + 2{\text{D tan}}\upvarphi_{\text{o}} } \right)^{2} \cot {{\upvarphi }}_{\text{o}}$$

(14)

$${\text{F}}_{{{\text{f}}1}} = {\text{m}}_{1} {\text{g}}\tan {{\upvarphi }}$$

(15)

$${\text{F}}_{{{\text{c}}1}} = {\text{C}}_{\text{o}} {\text{B}}\left( { {\text{H}} + 2{\text{D tan}}\upvarphi_{\text{o}} } \right)$$

(16)

$${\text{P}}_{{{\text{f}}1}} = \left( {{\text{F}}_{{{\text{f}}1}} + {\text{F}}_{{{\text{C}}1}} } \right)\tan {{\upvarphi }}$$

(17)

$${\text{P}}_{{{\text{c}}1}} = {\text{C}}_{\text{o}} {\text{BR}}\uptheta$$

(18)

$${\text{P}}_{\text{ad}} = {\text{A}}_{\text{d}} {\text{BR}}\uptheta$$

(19)

$${\text{P}}_{{{\text{f}}^{2}}} = \left( {{\text{F}}_{{{\text{f}}1}} + {\text{F}}_{{{\text{C}}1}} } \right)\tan {{\updelta }}$$

(20)

$${\text{m}}_{2} {\text{g}} = 2{{\upgamma }}_{\text{o}} {\text{BHD}}$$

(21)

$${\text{G}} = \frac{1}{6}\gamma{\text{D}}^{3} (1 - \sin {{\upvarphi }})\left( {\cot {{\upalpha }} + \cot {{\upbeta }}} \right)$$

(22)

$${\text{SF}}_{2 } = {\text{G tan}} \upvarphi$$

(23)

$${\text{CF}}_{2 } = \frac{1}{2}{\text{CD}}^{2} \left( {\cot {{\upalpha }} + \cot {{\upbeta }}} \right)$$

(24)

$${\text{m}}_{3} {\text{g}} = \frac{1}{2}\gamma{\text{BD}}^{2} \left( {\cot {{\upalpha }} + \cot {{\upbeta }}} \right)$$

(25)

$${\text{F}}_{\text{ad}} = \frac{{{\text{A}}_{\text{d}} }}{{\sin {{\upalpha }}}} {\text{BD}}$$

(26)

$${\text{W}} = {\text{P}}_{{{\text{f}}1}} + {\text{P}}_{{{\text{f}}2 }} + {\text{P}}_{\text{ad}} + {\text{m}}_{2} {\text{g}} + {\text{m}}_{3} {\text{g}}$$

(27)

$${\text{CF}}_{1} = \frac{\text{C}}{{\sin {{\upbeta }}}} {\text{BD}}$$

(28)

$${\text{SF}}_{1} = {\text{Q}}\tan {{\upvarphi }}$$

(29)

$${\text{P}}_{\text{r}} = \frac{{{\text{W}}\sin \left( {{{\upbeta }} + {{\upvarphi }}} \right) - {\text{F}}_{\text{ad}} \cos \left( {{{\upalpha }} + {{\upbeta }} + {{\upvarphi }}} \right) + 2 {\text{SF}}_{2} \cos \left( {{\upvarphi }} \right) + 2 {\text{CF}}_{2} \cos \left( {{\upvarphi }} \right) + {\text{CF}}_{1} \cos \left( {{\upvarphi }} \right)}}{{{ \sin }\left( {{{\upalpha }} + {{\upbeta }} + {{\upvarphi }} + {{\updelta }}} \right)}}$$

(30)

$${\text{F}}_{\text{x}} = {\text{P}}_{\text{r}} { \sin }\left( {{{\upalpha }} + {{\updelta }}} \right) + {\text{F }}_{{{\text{f}}1}} + {\text{F}}_{{{\text{c}}1}}$$

(31)

$${\text{F}}_{\text{y}} = {\text{P}}_{\text{r}} { \cos }\left( {{{\upalpha }} + {{\updelta }}} \right) - \left( {{\text{P }}_{{{\text{f}}2}} + {\text{P}}_{\text{ad}} } \right)$$

(32)

$${\text{F}} = \sqrt[2]{{{\text{F}}_{\text{x}}^{2} + {\text{F}}_{\text{y}}^{2} }}$$

(33)

Model of [2] Fundamental Equation of Earthmoving Mechanics. Nomenclature and description of equations can be found in [22]

$${\text{P}}_{\text{r}} = \left( {\varUpsilon {\text{gD}}^{2} {\text{N}}_{\varUpsilon } + {\text{CDN}}_{\text{c}} + {\text{qDN}}_{\text{q}} + {\text{A}}_{\text{d}} {\text{DN}}_{\text{Ad }} + \varUpsilon {\text{v}}^{2} {\text{DN}}_{\text{a}} } \right){\text{B}}$$

(34)

$${\text{N}}_{{{\upgamma }}} = \frac{{{\text{cot} \upalpha } + {\text{cot}} \upbeta }}{{2\left[ {{ \cos }\left( {{{\upalpha }} + {{\updelta }}} \right) + {\text{sin}}\left( {{{\upalpha }} + {{\updelta }}} \right) {\text{cot}}\left( {{{\upbeta }} + {{\upvarphi }}} \right)} \right]}}$$

(35)

$${\text{N}}_{\text{c}} = \frac{{{\text{cot} \upalpha } + {\text{cot}} \upbeta }}{{2\left[ {{ \cos }\left( {{{\upalpha }} + {{\updelta }}} \right) + {\text{sin}}\left( {{{\upalpha }} + {{\updelta }}} \right){ \cot }\left( {{{\upbeta }} + {{\upvarphi }}} \right)} \right]}}$$

(36)

$${\text{N}}_{\text{q}} = \frac{{{\text{cot} \upalpha } + {\text{cot}} \upbeta }}{{{ \cos }\left( {{{\upalpha }} + {{\updelta }}} \right) + {\text{sin}}\left( {{{\upalpha }} + {{\updelta }}} \right) {\text{cot}}\left( {{{\upbeta }} + \upphi } \right)}}$$

(37)

$${\text{q}} = \frac{{{\text{P}}_{{{\text{f}}1}} + {\text{P}}_{{{\text{f}}2}} + {\text{P}}_{{{\text{c}}1}} + {\text{m}}_{2} {\text{g}}}}{{2{\text{BD}}}}$$

(38)

$${\text{F }} = {\text{P}}_{\text{r}} {\text{sin }}\left( {{{\upalpha }} + {{\updelta }}} \right)$$

(39)