Engineering Problems

F1(x::Vector{Float64}) -> Float64

Tension/Compression Spring Design Optimization.

Minimizes the weight of a tension/compression spring subject to constraints on shear stress, surge frequency, minimum deflection, and geometric limits.

Problem Source

A well-known benchmark in constrained engineering design, commonly used in metaheuristic optimization literature.

Variables

• x[1]: Wire diameter (d)
• x[2]: Mean coil diameter (D)
• x[3]: Number of active coils (N)

Constraints

Four nonlinear inequality constraints:

• Shear stress constraint
• Surge frequency constraint
• Minimum deflection constraint
• Geometry-related limits

Returns

• Penalized objective function value (Float64)

F2(x::Vector{Float64}) -> Float64

Pressure Vessel Design Optimization.

Minimizes the total cost of a cylindrical pressure vessel, which includes material, forming, and welding costs, subject to constraints on thickness, volume, and stress.

Problem Source

A classical benchmark problem in constrained engineering design, widely used in metaheuristic algorithm evaluations.

Variables

• x[1]: Thickness of the shell (Ts)
• x[2]: Thickness of the head (Th)
• x[3]: Inner radius (R)
• x[4]: Length of the cylindrical section without head (L)

Constraints

Four nonlinear inequality constraints:

• Stress constraints on thickness
• Volume constraint
• Geometrical bounds

Returns

• Penalized objective function value (Float64)

Engineering_F3(x::Vector{Float64}) -> Float64

Welded Beam Design Optimization Problem.

Minimizes the cost of a welded beam subject to constraints on shear stress, normal stress, deflection, and geometric properties.

Objective

\[\vec{z} = [z_1, z_2, z_3, z_4] = [h, l, t, b] \\ min_{\vec{z}} f(\vec{z}) = 1.10471 z_1^2 z_2 + 0.04811 z_3 z_4 (14 + z_2)\]

Constraints

\[\begin{aligned} g_1(\vec{z}) &= \tau(z) - \tau_{\max} \leq 0 \\ g_2(\vec{z}) &= \sigma(z) - \sigma_{\max} \leq 0 \\ g_3(\vec{z}) &= z_1 - z_4 \leq 0 \\ g_4(\vec{z}) &= 0.10471 z_1^2 + 0.04811 z_3 z_4 (14 + z_2) - 5 \leq 0 \\ g_5(\vec{z}) &= 0.125 - z_1 \leq 0 \\ g_6(\vec{z}) &= \delta(z) - \delta_{\max} \leq 0 \\ g_7(\vec{z}) &= P - P_c(z) \leq 0 \end{aligned}\]

Definitions

\[\tau(z) = \sqrt{(\tau')^2 + 2\tau'\tau''\frac{z_2}{2R} + (\tau'')^2},\quad \tau' = \frac{P}{\sqrt{2} z_1 z_2},\quad \tau'' = \frac{MR}{J}\]

\[M = P \left( L + \frac{z_2}{2} \right),\quad R = \sqrt{ \frac{z_2^2}{4} + \left( \frac{z_1 + z_3}{2} \right)^2 }\]

\[J = 2 \sqrt{2} z_1 z_2 \left[ \frac{z_2^2}{12} + \left( \frac{z_1 + z_3}{2} \right)^2 \right]\]

\[\sigma(z) = \frac{6PL}{z_4 z_3^2},\quad \delta(z) = \frac{4PL^3}{E z_3^3 z_4}\]

\[P_c(z) = \frac{4.013 E \sqrt{z_3^2 z_4^5 / 36}}{L^2} \left( 1 - \frac{z_3}{2L} \sqrt{\frac{E}{4G}} \right)\]

Constants

• P = 6000 lb
• L = 14 in
• E = 30×10⁶ psi
• G = 12×10⁶ psi
• τₘₐₓ = 13600 psi
• σₘₐₓ = 30000 psi
• δₘₐₓ = 0.25 in

Decision Variables

• x[1] = z₁: Thickness of weld (h)
• x[2] = z₂: Length of weld (l)
• x[3] = z₃: Height of beam (t)
• x[4] = z₄: Width of beam (b)

Returns

• Penalized objective function value (Float64)

F4(x::Vector{Float64}) -> Float64

Speed Reducer Design Optimization.

Minimizes the weight of a speed reducer subject to constraints on bending stress, surface stress, transverse deflections, and geometry.

Problem Source

A standard benchmark problem in engineering design, commonly used to test constrained optimization algorithms.

Variables

• x[1]: Face width (in)
• x[2]: Module of teeth (in)
• x[3]: Number of teeth
• x[4]: Length of the first shaft between bearings (in)
• x[5]: Length of the second shaft between bearings (in)
• x[6]: Diameter of the first shaft (in)
• x[7]: Diameter of the second shaft (in)

Constraints

• Bending stress
• Surface stress
• Deflection of shafts
• Geometric and design constraints
• Seven nonlinear inequality constraints in total

Returns

• Penalized objective function value (Float64)

F5(x::Vector{Float64}) -> Float64

Gear Train Design Optimization.

Minimizes the error between an actual and a desired gear ratio in a simple four-gear train. All variables must be integers.

Problem Source

A discrete constrained engineering design problem widely used to evaluate optimization algorithms that handle integer variables.

Variables

• x[1]: Number of teeth on gear 1 (integer)
• x[2]: Number of teeth on gear 2 (integer)
• x[3]: Number of teeth on gear 3 (integer)
• x[4]: Number of teeth on gear 4 (integer)

Constraints

• Each variable must be an integer in the range [12, 60]
• The gear ratio error must be minimized

Returns

• Squared error between actual and desired gear ratio (Float64)

F6(x::Vector{Float64}) -> Float64

Three-Bar Truss Design Optimization.

Minimizes the weight of a three-bar truss structure subject to stress and displacement constraints.

Problem Source

A classical structural optimization benchmark problem used in metaheuristic algorithm research.

Variables

• x[1]: Cross-sectional area of the first bar (continuous)
• x[2]: Cross-sectional area of the second bar (continuous)

Constraints

• Stress in each member must not exceed allowable limits
• Displacement constraints on the structure
• Variable bounds typically in the range [0.1, 10]

Returns

• Penalized objective function value (Float64) representing the weight of the truss

F7(x::Vector{Float64}) -> Float64

Rolling Element Bearing Design Optimization.

Minimizes the bearing’s weight subject to constraints on stress, deflection, and geometry.

Problem Source

A standard constrained engineering design problem often used to benchmark metaheuristic algorithms.

Variables

• x[1]: Bearing inner radius
• x[2]: Bearing outer radius
• x[3]: Width of the bearing
• x[4]: Shaft diameter
• x[5]: Number of rolling elements

Constraints

• Stress limits on the bearing components
• Deflection limits
• Geometric and manufacturing constraints

Returns

• Penalized objective function value (Float64) reflecting the bearing weight or cost

F8(x::Vector{Float64}) -> Float64

Cantilever Beam Design Optimization.

Minimizes the weight of a cantilever beam subject to constraints on bending stress, deflection, and geometric dimensions.

Problem Source

A classical constrained engineering design problem used in metaheuristic algorithm research.

Variables

• x[1]: Width of the beam cross-section
• x[2]: Height of the beam cross-section
• x[3]: Length of the beam segment 1
• x[4]: Length of the beam segment 2
• x[5]: Length of the beam segment 3
• x[6]: Length of the beam segment 4

Constraints

• Maximum bending stress constraints
• Deflection limits at the beam’s free end
• Geometric bounds on variables

Returns

• Penalized objective function value (Float64), representing the beam weight

F9(x::Vector{Float64}) -> Float64

I-Beam Deflection Optimization.

Minimizes the weight of an I-beam subject to constraints on bending stress, shear stress, and deflection under load.

Problem Source

A classical engineering design benchmark widely used in metaheuristic algorithm literature.

Variables

• x[1]: Web height
• x[2]: Flange width
• x[3]: Web thickness
• x[4]: Flange thickness

Constraints

• Bending stress limits
• Shear stress limits
• Maximum deflection allowed
• Geometric constraints

Returns

• Penalized objective function value (Float64), representing the beam weight