You are an expert in reformulating **linear fractional terms (expressions of form $y/x, x \neq 0$)** into purely linear constraints for mixed-integer optimization problems.

## TASK

For each **linear fractional term $y/x$** in the LaTeX optimization problem, complete steps A–D clearly and concisely:

**A IDENTIFY** – Quote the exact linear quotient terms and its index set.

**B EVALUATE** – Check these methods (**stop at first Applicable=YES, Exact=YES**):

1. **Charnes-Cooper Transformation**
2. **Homogenisation / Normalisation**

Briefly summarize evaluation (e.g., Direct: Applicable=…, Exact=…).

**C FORMULATE** – Provide constraints explicitly in LaTeX; no other changes.

**D VERIFY** – One concise sentence explicitly confirming no nonlinearities or reciprocals remain.

**CRITICAL:**

**DO NOT replace one linear fractional term with another.**

**DO NOT leave any nonlinear terms (e.g. bilinear terms) in the optimization problem where variables are involved.**


## INPUT

* **LaTeX model:** {latex_model}
* **Pattern description (human hint):** {quotient_pattern}
* **Concrete parameters (bounds, indices):** {param_info}

## OUTPUT FORMAT

REPORT:
Pattern      : <exact quotient expression and indices>
Technique    : <chosen method>
Verification : <one concise sentence explicitly stating linearity>
Bounds / M   : <bounds or Big-M; "n/a" if not used>
Aux vars     : <any new variables; minimal>

## UPDATED MODEL

<full reformulated LaTeX model>

**IMPORTANT:**

* Reformulated model must be purely linear (LP/MILP).
* **NO nonlinear terms or reciprocals remain.**
* Prefer simplest methods without binaries if possible.
