Podcast
Questions and Answers
Which manufacturing process involves shaping liquid or malleable raw material using a frame?
Which manufacturing process involves shaping liquid or malleable raw material using a frame?
- Moulding (correct)
- Extrusion
- Turning
- Laminating
Which process creates objects by pushing material through a die?
Which process creates objects by pushing material through a die?
- Extrusion (correct)
- Turning
- Stamping
- Casting
What is the technique of manufacturing materials in multiple layers?
What is the technique of manufacturing materials in multiple layers?
- Turning
- Milling
- Casting
- Laminating (correct)
Which manufacturing process involves using rotary cutters to remove material from a workpiece?
Which manufacturing process involves using rotary cutters to remove material from a workpiece?
Which process involves rotating a workpiece and using a cutting tool?
Which process involves rotating a workpiece and using a cutting tool?
What process involves pouring liquid into a mold to solidify?
What process involves pouring liquid into a mold to solidify?
Which process uses a press and die to shape metal?
Which process uses a press and die to shape metal?
What type of manufacturing involves permanently deforming a material?
What type of manufacturing involves permanently deforming a material?
During injection molding, what form are the plastics when fed into the hopper?
During injection molding, what form are the plastics when fed into the hopper?
In injection molding, what happens to the molten plastic after it is injected into the mold cavity?
In injection molding, what happens to the molten plastic after it is injected into the mold cavity?
What is being created when a heater polymer tube is inflated to take the shape of a hollow form with a thin wall.
What is being created when a heater polymer tube is inflated to take the shape of a hollow form with a thin wall.
What process involves decomposing organic materials into simpler substances?
What process involves decomposing organic materials into simpler substances?
What is created when materials are heated and their shape is altered?
What is created when materials are heated and their shape is altered?
What is a jig used for during production?
What is a jig used for during production?
What is the purpose of a fixture during production?
What is the purpose of a fixture during production?
What is a pre-production prototype considered?
What is a pre-production prototype considered?
What is the goal of tooling up a production line?
What is the goal of tooling up a production line?
What is carbon fiber primarily used for in racing bicycles?
What is carbon fiber primarily used for in racing bicycles?
What is a common use for Kevlar material?
What is a common use for Kevlar material?
What is the typical size range of nanomaterials?
What is the typical size range of nanomaterials?
Flashcards
Injection moulding
Injection moulding
A production process employed during the mass production of components made from polymers. Granular form pellets are fed into a hopper, and gravity is fed into a heater where they become molten.
Jig
Jig
A jig is a device used to guide a tool to a specific position on a workpiece. Jigs help speed up the making process and can eliminate errors ensuring every operation is repeated exactly the same.
Fixture
Fixture
A fixture is a tool which holds the workpiece to a machine bed, which will allow a cutting or shaping operation to occur without the need to measure, mark or move the workpiece.
Prototype
Prototype
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Modern materials
Modern materials
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Carbon fibre
Carbon fibre
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Kevlar
Kevlar
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SMART materials
SMART materials
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Nanomaterials
Nanomaterials
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Titanium
Titanium
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Shape memory alloys (SMAs)
Shape memory alloys (SMAs)
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Elasticity
Elasticity
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Durability
Durability
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Tensile strength
Tensile strength
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Compressive strength
Compressive strength
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Malleability
Malleability
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Plating
Plating
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Heat treatments
Heat treatments
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Stains
Stains
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Joining or Forming
Joining or Forming
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Study Notes
Duality in Linear Programming
- Examines the relationship between a primal and a dual linear programming (LP) problem.
Primal Problem (Maximization)
- The primal LP seeks to maximize $c^Tx$.
- Subject to constraints $Ax \le b$ and $x \ge 0$.
Dual Problem (Minimization)
- The dual LP seeks to minimize $b^Ty$.
- Subject to constraints $A^Ty \ge c$ and $y \ge 0$.
Duality Theorems
Weak Duality Theorem
- If $x$ is a feasible solution to the primal LP and $y$ is a feasible solution to the dual LP, the objective values satisfy $c^Tx \le b^Ty$.
Strong Duality Theorem
- If the primal LP has an optimal solution $x^$, then the dual LP also has an optimal solution $y^$, such that their objective values are equal: $c^Tx^* = b^Ty^*$.
Complementary Slackness Theorem
- Relates optimal primal and dual solutions.
- At optimality, for all $i$, $y_i^* (b_i - a_i^T x^) = 0$, meaning if $y_i^ > 0$, then the $i$-th constraint in the primal problem is tight ($a_i^T x^* = b_i$), and vice versa.
- At optimality, for all $j$, $x_j^* (a_j^T y^* - c_j) = 0$, meaning if $x_j^* > 0$, then the $j$-th constraint in the dual problem is tight ($a_j^T y^* = c_j$), and vice versa.
Economic Interpretation of the Dual Problem
- The primal LP can represent a profit-maximization problem with limited resources.
- $x_j$ refers to the amount of activity $j$ to perform.
- $c_j$ is the profit per unit of activity $j$.
- $b_i$ is the amount of resource $i$ that is available.
- $a_{ij}$ is the amount of resource $i$ used by activity $j$.
- The dual variable $y_i$ represents the shadow price(or imputed value) of resource $i$.
- Key Idea: The dual minimizes imputed value of all resources s.t. imputed cost of inputs for each activity $j$ is no less than its profit $c_j$
Shadow Prices
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Shadow price are how much the objective function value would change with a one-unit increase in the availability of the resource.
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Illustrative Example:
-
A maximization problem with constraints:
- $x_1 + x_2 + x_3 \le 100$
- $2x_1 + x_2 + 3x_3 \le 55$
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Corresponding dual variables $y_1$ and $y_2$.
-
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$y_1$ could be interpreted as the increase in profit per unit increase in resource 1.
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$y_2$ could be interpreted as the increase in profit per unit increase in resource 2.
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