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Equation 1 · Chiplets and Advanced Packaging in Practice: Floorplanning, UCIe, and Thermal Co-Design

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Cost(P)  =  α⋅WL(P)  +  β⋅Tmax⁡(P)  +  γ⋅σmax⁡(P),\text{Cost}(\mathcal{P}) \;=\; \alpha \cdot WL(\mathcal{P}) \;+\; \beta \cdot T_{\max}(\mathcal{P}) \;+\; \gamma \cdot \sigma_{\max}(\mathcal{P}),

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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P\mathcal{P}

Symbol P

a candidate placement of dies on the carrier, WL is total interconnect wirelength.

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α\alpha

Symbol α

the whichever of.

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WW

Symbol W

W appears in the objective or constraint used by the optimization on the right.

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LL

Symbol L

L appears in the objective or constraint used by the optimization on the right.

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β\beta

Symbol β

β appears in the objective or constraint used by the optimization on the right.

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Tmax⁡T_{\max}

Symbol T_max

peak steady-state junction temperature anywhere in the assembly.

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γ\gamma

Symbol gamma

the design team weights most heavily determines where dies actually land.

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σmax⁡\sigma_{\max}

Symbol sigma_max

peak mechanical stress at any bonded interface.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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multiplication

multiplication

Multiply the quantities on either side.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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What the article says around this equation

A floorplanning objective built to hold all three properties at once is naturally written as a weighted combination rather than a single term: Cost(P)  =  α⋅WL(P)  +  β⋅Tmax⁡(P)  +  γ⋅σmax⁡(P)\text{Cost}(\mathcal{P}) \;=\; \alpha \cdot WL(\mathcal{P}) \;+\; \beta \cdot T_{\max}(\mathcal{P}) \;+\; \gamma \cdot \sigma_{\max}(\mathcal{P}). where P\mathcal{P} is a candidate placement of dies on the carrier, WL is total interconnect wirelength, Tmax⁡T_{\max} is peak steady-state junction temperature anywhere in the assembly, and σmax⁡\sigma_{\max} is peak mechanical stress at any bonded interface. This is a schematic statement of the shape these tools optimize, not a reproduction of any one paper’s exact formula, but it makes the practitioner point concrete: whichever of α\alpha , β\beta , and γ\gamma a design team weights most heavily determines where dies actually land, and a…
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A floorplanning objective built to hold all three properties at once is naturally written as a weighted combination rather than a single term: Cost(P)  =  α⋅WL(P)  +  β⋅Tmax⁡(P)  +  γ⋅σmax⁡(P)\text{Cost}(\mathcal{P}) \;=\; \alpha \cdot WL(\mathcal{P}) \;+\; \beta \cdot T_{\max}(\mathcal{P}) \;+\; \gamma \cdot \sigma_{\max}(\mathcal{P}). where P\mathcal{P} is a candidate placement of dies on the carrier, WL is total interconnect wirelength, Tmax⁡T_{\max} is peak steady-state junction temperature anywhere in the assembly, and σmax⁡\sigma_{\max} is peak mechanical stress at any bonded interface. This is a schematic statement of the shape these tools optimize, not a reproduction of any one paper’s exact formula, but it makes the practitioner point concrete: whichever of α\alpha , β\beta , and γ\gamma a design team weights most heavily determines where dies actually land, and a floorplan tuned only for WL is optimizing a real but incomplete objective.

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