cuopt-numerical-optimization-formulation

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Numerical Optimization Formulation

数值优化建模

Concepts and workflow for going from a problem description to a clear formulation across LP, MILP, and QP. No API code here.

从问题描述到LP、MILP、QP清晰建模的概念与工作流程。本文不包含API代码。

What is LP / MILP / QP

什么是LP / MILP / QP

LP: Linear objective, linear constraints, continuous variables.
MILP: Same as LP plus some integer or binary variables (e.g., scheduling, facility location, selection).
QP: Quadratic objective (e.g., x², x·y terms — portfolio variance, least squares), linear constraints. QP support in cuOpt is currently in beta.

LP：线性目标函数、线性约束、连续变量。
MILP：与LP相同，额外包含部分整数或二进制变量（例如调度、设施选址、选择问题）。
QP：二次目标函数（例如x²、x·y项——投资组合方差、最小二乘法）、线性约束。cuOpt中的QP支持目前处于beta阶段。

Identifying problem type

问题类型识别

Property	LP	MILP	QP
Objective	Linear	Linear	Quadratic (xᵀQx + cᵀx)
Constraints	Linear	Linear	Linear (no quadratic constraints)
Variables	Continuous	Mixed: continuous + integer/binary	Continuous
Sense	min or max	min or max	minimize only (negate to max)

If the objective is purely linear, prefer LP/MILP — do not artificially introduce quadratic terms. If any variable is integer or binary, the problem is MILP regardless of the rest.

属性	LP	MILP	QP
目标函数	线性	线性	二次型（xᵀQx + cᵀx）
约束	线性	线性	线性（不支持二次约束）
变量	连续	混合：连续 + 整数/二进制	连续
优化方向	最小化或最大化	最小化或最大化	仅支持最小化（如需最大化，取目标函数负值后再最小化）

若目标函数完全为线性，优先选择LP/MILP——不要人为引入二次项。若存在任何整数或二进制变量，无论其他属性如何，该问题均为MILP。

Required formulation questions

建模必问问题

Ask these if not already clear:

Decision variables — What are they? Bounds?
Objective — Minimize or maximize? Linear or quadratic? For QP: any squared or cross terms (x², x·y)? If maximize a quadratic, the user must negate and minimize.
Constraints — Linear inequalities/equalities? (Quadratic constraints are not supported.)
Variable types — All continuous (LP / QP) or some integer/binary (MILP)?
Convexity (QP only) — For minimization, the quadratic form (matrix Q) should be positive semi-definite for well-posed problems.

若信息不明确，需询问以下问题：

决策变量——变量是什么？取值范围？
目标函数——最小化还是最大化？线性还是二次？对于QP：是否包含平方项或交叉项（x²、x·y）？若最大化二次函数，用户必须取其负值后进行最小化。
约束条件——线性不等式/等式？（不支持二次约束）
变量类型——全部为连续变量（LP / QP）还是包含部分整数/二进制变量（MILP）？
凸性（仅QP）——对于最小化问题，二次型矩阵Q应半正定，以保证问题适定。

Typical modeling elements

典型建模元素

Continuous variables — production amounts, flow, allocations, portfolio weights.
Binary variables — open/close, yes/no (e.g., facility open, item selected).
Linking constraints — e.g., production only if facility open (Big-M or indicator).
Resource constraints — linear cap on usage (materials, time, capacity).
Quadratic objective terms — variance (xᵀQx), squared error (‖Ax − b‖²), interaction terms.

连续变量——产量、流量、分配量、投资组合权重。
二进制变量——开启/关闭、是/否（例如设施是否启用、是否选中某物品）。
关联约束——例如仅当设施启用时才可生产（Big-M法或指示变量法）。
资源约束——对资源使用的线性上限（原材料、时间、产能）。
二次目标项——方差（xᵀQx）、平方误差（‖Ax − b‖²）、交互项。

Typical QP use cases

QP典型应用场景

Portfolio optimization — minimize variance subject to return and budget.
Least squares — minimize ‖Ax − b‖² subject to linear constraints.
Other quadratic objectives with linear constraints.

投资组合优化——在收益和预算约束下最小化方差。
最小二乘法——在线性约束下最小化‖Ax − b‖²。
其他带有线性约束的二次目标问题。

Problem statement parsing

问题描述解析

When the user gives problem text, classify every sentence and then summarize before formulating. The parsing framework below applies regardless of LP / MILP / QP.

Classify every sentence as parameter/given, constraint, decision, or objective. Watch for implicit constraints (e.g., committed vs optional phrasing) and implicit objectives (e.g., "determine the plan" + costs → minimize total cost).

Ambiguity: If anything is still ambiguous, ask the user or solve all plausible interpretations and report all outcomes; do not assume a single interpretation.

当用户提供问题文本时，需对每个句子进行分类，然后总结后再建模。以下解析框架适用于所有LP / MILP / QP问题。

将每个句子分类为参数/给定条件、约束、决策或目标。注意识别隐含约束（例如承诺性与可选性表述）和隐含目标（例如“制定计划”+成本信息→最小化总成本）。

歧义处理：若存在任何歧义，需询问用户或求解所有合理的解释并报告所有结果；不要默认单一解释。

🔒 MANDATORY: When in Doubt — Ask

🔒 强制要求：存疑必问

If there is any doubt about whether a constraint or value should be included, ask the user and state the possible interpretations.

若对是否应包含某约束或数值存在任何疑问，务必询问用户并说明可能的解释。

🔒 MANDATORY: Complete-Path Runs — Try All Variants

🔒 强制要求：全路径运行——尝试所有变体

When the user asks to run the complete path (e.g., end-to-end, full pipeline), run all plausible variants and report all outcomes so the user can choose; do not assume a single interpretation.

当用户要求运行完整流程（例如端到端、全流水线）时，需运行所有合理的变体并报告所有结果供用户选择；不要默认单一解释。

Three labels

三类标签

Label	Meaning	Examples (sentence type)
Parameter / given	Fixed data, inputs, facts. Not chosen by the model.	"Demand is 100 units." "There are 3 factories." "Costs are $5 per unit."
Constraint	Something that must hold. May be explicit or implicit from phrasing.	"Capacity is 200." "All demand must be met." "At least 2 shifts must be staffed."
Decision	Something we choose or optimize.	"How much to produce." "Which facilities to open." "How many workers to hire."
Objective	What to minimize or maximize. May be explicit ("minimize cost") or implicit ("determine the plan" with costs given).	"Minimize total cost." "Determine the production plan" (with costs) → minimize total cost.

标签	含义	示例（句子类型）
参数/给定条件	固定数据、输入、事实。不由模型选择。	“需求为100单位。” “有3家工厂。” “单位成本为5美元。”
约束	必须满足的条件。可能是显式的，也可能是表述中的隐含条件。	“产能为200。” “必须满足所有需求。” “至少需配备2班人员。”
决策	我们需要选择或优化的内容。	“生产多少数量。” “启用哪些设施。” “雇佣多少工人。”
目标	需要最小化或最大化的内容。可能是显式（“最小化成本”）或隐含（“制定计划”+给定成本信息）。	“最小化总成本。” “制定生产计划”（给定成本）→ 最小化总成本。

Implicit constraints: committed vs optional phrasing

隐含约束：承诺性与可选性表述

Committed/fixed phrasing → treat as parameter or implicit constraint (everything mentioned is given or must happen). Not a decision.

Phrasing	Interpretation	Why
"Plans to produce X products"	Constraint: all X must be produced.	Commitment; production level is fixed.
"Operates 3 factories"	Parameter: all 3 are open. Not a location-selection problem.	Current state is fixed.
"Employs N workers"	Parameter: all N are employed. Not a hiring decision.	Workforce size is given.
"Has a capacity of C"	Parameter (C) + constraint: usage ≤ C.	Capacity is fixed.
"Must meet all demand"	Constraint: demand satisfaction.	Explicit requirement.

Optional/decision phrasing → treat as decision.

Phrasing	Interpretation	Why
"May produce up to …"	Decision: how much to produce.	Optional level.
"Can choose to open" (factories, sites)	Decision: which to open.	Selection is decided.
"Considers hiring"	Decision: how many to hire.	Hiring is under consideration.
"Decides how much to order"	Decision: order quantities.	Explicit decision.
"Wants to minimize/maximize …"	Objective (drives decisions).	Goal; decisions are the levers.

承诺性/固定表述 → 视为参数或隐含约束（提及的所有内容均为给定条件或必须执行）。不属于决策范畴。

表述	解读	原因
“计划生产X种产品”	约束：必须生产全部X种产品。	属于承诺；产量固定。
“运营3家工厂”	参数：3家工厂均已启用。不属于选址问题。	当前状态固定。
“雇佣N名工人”	参数：已雇佣全部N名工人。不属于招聘决策问题。	员工规模给定。
“产能为C”	参数（C） + 约束：使用量 ≤ C。	产能固定。
“必须满足所有需求”	约束：需求必须被满足。	显式要求。

可选性/决策表述 → 视为决策。

表述	解读	原因
“最多可生产……”	决策：生产数量由决策决定。	可选产量。
“可选择启用”（工厂、站点）	决策：选择启用哪些设施。	需决定选择对象。
“考虑雇佣”	决策：雇佣人数由决策决定。	雇佣处于考量阶段。
“决定订购数量”	决策：订购数量由决策决定。	显式决策。
“希望最小化/最大化……”	目标（驱动决策）。	目标；决策是实现目标的手段。

Implicit objectives — do not miss

隐含目标——不可遗漏

If the problem asks to "determine the plan" (or similar) but does not state "minimize" or "maximize" explicitly, the objective is often implicit. You MUST identify it and state it before formulating; do not build a model with no objective.

Phrasing / context	Likely implicit objective	Why
"Determine the production plan" + costs given (per unit, per hour, etc.)	Minimize total cost (production + inspection/sales + overtime, etc.)	Plan is chosen; costs are specified → natural goal is to minimize total cost.
"Determine the plan" + costs and revenues given	Maximize profit (revenue − cost)	Both sides of the ledger → optimize profit.
"Try to determine the monthly production plan" + workshop hour costs, inspection/sales costs	Minimize total cost	All cost components are given; no revenue to maximize → minimize total cost.

Rule: When the problem gives cost (or cost and revenue) data and asks to "determine", "find", or "establish" the plan, always state the objective explicitly (e.g., "I'm treating the objective as minimize total cost, since only costs are given."). If both cost and revenue are present, state whether you use "minimize cost" or "maximize profit". Ask the user if unclear.

若问题要求“制定计划”（或类似表述）但未明确说明“最小化”或“最大化”，目标往往是隐含的。 你必须先识别并明确目标再建模；不要构建无目标的模型。

表述/上下文	可能的隐含目标	原因
“制定生产计划”+给定成本信息（单位成本、小时成本等）	最小化总成本（生产成本+检验/销售成本+加班成本等）	计划需选定；成本已明确→自然目标是最小化总成本。
“制定计划”+给定成本和收入信息	最大化利润（收入−成本）	收支均已明确→优化利润。
“尝试制定月度生产计划”+车间小时成本、检验/销售成本	最小化总成本	仅给定成本构成；无收入需最大化→最小化总成本。

规则：当问题提供成本（或成本与收入）数据并要求“制定”、“找出”或“确立”计划时，务必明确说明目标（例如“由于仅提供了成本信息，我将目标视为最小化总成本。”）。若同时提供成本和收入，需说明是选择“最小化成本”还是“最大化利润”。若不明确，询问用户。

Parsing workflow

解析流程

Split the problem text into sentences or logical clauses.
Label each: parameter/given | constraint | decision | objective (if stated).
Identify the objective (explicit or implicit): If the problem says "minimize/maximize X", that's the objective. If it only says "determine the plan" (or "find", "establish") but gives costs (and possibly revenues), the objective is implicit — state it (e.g., minimize total cost, or maximize profit) and confirm with the user if ambiguous.
Flag implicit constraints: For each sentence, ask — "Does this state a fixed fact or a requirement (→ parameter/constraint), or something we choose (→ decision)?"
Resolve ambiguity by checking verbs and modals:
- "is", "has", "operates", "employs", "plans to" (fixed/committed) → parameter or implicit constraint.
- "may", "can choose", "considers", "decides", "wants to" (optional) → decision or objective.
🔒 MANDATORY — If anything is still ambiguous (e.g., a value or constraint could be read two ways): ask the user which interpretation is correct, or solve all plausible interpretations and report all outcomes. Do not assume a single interpretation.
Summarize for the user: list parameters, constraints (explicit + flagged implicit), decisions, and objective (explicit or inferred) before writing the math formulation.

拆分：将问题文本拆分为句子或逻辑分句。
标记：为每个分句标记：参数/给定条件 | 约束 | 决策 | 目标（若明确说明）。
识别目标（显式或隐含）：若问题说明“最小化/最大化X”，即为目标。若仅说明“制定计划”（或“找出”、“确立”）但提供了成本（可能包含收入）信息，目标为隐含——需明确说明（例如最小化总成本或最大化利润），若存在歧义需与用户确认。
标记隐含约束：对每个句子，自问——“该句子表述的是固定事实或要求（→参数/约束），还是我们需选择的内容（→决策）？”
通过动词和情态动词解决歧义：
- “is”、“has”、“operates”、“employs”、“plans to”（固定/承诺性）→ 参数或隐含约束。
- “may”、“can choose”、“considers”、“decides”、“wants to”（可选性）→ 决策或目标。
🔒 强制要求——若仍存在歧义（例如某数值或约束有两种解读方式）：询问用户哪种解读正确，或求解所有合理的解释并报告所有结果。不要默认单一解释。
总结：在撰写数学建模公式前，为用户列出参数、约束（显式+标记的隐含约束）、决策以及目标（显式或推断得出）。

Parsing checklist

解析检查清单

Example

示例

Text: "The company operates 3 factories and plans to produce 500 units. It may use overtime at extra cost. Minimize total cost."

Sentence / phrase	Label	Note
"Operates 3 factories"	Parameter	All 3 open; not facility selection.
"Plans to produce 500 units"	Constraint (implicit)	All 500 must be produced.
"May use overtime at extra cost"	Decision	How much overtime is a decision.
"Minimize total cost"	Objective	Drives decisions.

Result: Parameters = 3 factories, 500 units target. Constraints = produce exactly 500 (implicit from "plans to produce"). Decisions = production allocation across factories, overtime amounts. Objective = minimize cost.

Implicit-objective example: A problem that asks to "determine the production plan" (or similar) and gives cost components (e.g., workshop, inspection, sales) but does not state "minimize" or "maximize" → Objective is implicit: minimize total cost. Always state it explicitly: "The objective is to minimize total cost."

文本：“公司运营3家工厂，计划生产500单位产品。可使用加班但需额外付费。最小化总成本。”

句子/短语	标签	说明
“运营3家工厂”	参数	3家工厂均已启用；不属于设施选址问题。
“计划生产500单位产品”	约束（隐含）	必须生产全部500单位产品。
“可使用加班但需额外付费”	决策	加班时长由决策决定。
“最小化总成本”	目标	驱动决策。

结果：参数=3家工厂、500单位生产目标。约束=必须生产恰好500单位（来自“计划生产”的隐含约束）。决策=各工厂生产分配量、加班时长。目标=最小化成本。

隐含目标示例：问题要求“制定生产计划”（或类似表述）并提供成本构成（例如车间、检验、销售成本）但未说明“最小化”或“最大化”→ 目标为隐含：最小化总成本。务必明确说明：“目标为最小化总成本。”

QP rule: minimize only

QP规则：仅支持最小化

QP objectives must be minimization. To maximize a quadratic expression, negate it and minimize; then negate the optimal value.

For minimization to be well-posed, the quadratic form

should be positive semi-definite. If

is indefinite, the problem is non-convex and may not have a finite optimum.

QP目标函数必须为最小化。如需最大化二次表达式，取其负值后进行最小化；之后再将最优值取负值。

为保证最小化问题适定，二次型

应半正定。若

不定，问题为非凸型，可能不存在有限最优解。

Common patterns

常见建模模式

The remaining sections cover specific LP/MILP modeling patterns. Each is independent — read the one that matches your problem.

剩余章节涵盖特定LP/MILP建模模式。各章节独立——阅读与你的问题匹配的章节即可。

Piecewise-linear objectives with integer production

带整数产量的分段线性目标函数

When modeling concave piecewise-linear profit/cost functions (e.g., decreasing marginal profit for bulk sales), the standard approach uses continuous segment variables with upper bounds equal to each segment's width. For a maximization with concave profit, the solver fills higher-profit segments first naturally.

Gotcha: If the quantity being produced is discrete (pieces, units, items), the total production variable must be INTEGER, even though segment variables can remain CONTINUOUS. Without this, the LP relaxation may yield a fractional total that produces a different (higher or lower) objective than the true integer optimum.

当建模凹分段线性利润/成本函数（例如批量销售的边际利润递减）时，标准方法使用连续分段变量，其上限等于各分段的宽度。对于凹利润的最大化问题，求解器会自然优先填充高利润分段。

注意事项：若生产数量为离散型（件、单位、物品），总产量变量必须为整数，即使分段变量可保持连续。否则，LP松弛可能得到分数形式的总产量，其目标值与真实整数最优解不同（偏高或偏低）。

Pattern

模式

x_total  — INTEGER (total production of a product)
s1, s2, … — CONTINUOUS (amount sold in each price segment, bounded by segment width)

Link: x_total = s1 + s2 + …
Resource constraints use x_total.
Objective uses segment variables × segment profit rates.

x_total  — INTEGER (total production of a product)
s1, s2, … — CONTINUOUS (amount sold in each price segment, bounded by segment width)

Link: x_total = s1 + s2 + …
Resource constraints use x_total.
Objective uses segment variables × segment profit rates.

Cutting stock / trim loss problems

下料/切料损耗问题

In cutting stock problems, waste area includes both trim loss (unused width within each cutting pattern) and over-production (excess strips produced beyond demand). Minimizing only trim loss (waste width × length per pattern) ignores over-production and yields an incorrect objective.

在下料问题中，废料面积包括切料损耗（每个下料模式内未使用的宽度）和过度生产（超出需求的条料产量）。仅最小化切料损耗（废料宽度×每个模式的长度）会忽略过度生产，导致目标函数错误。

Correct objective

正确目标函数

Since the total useful area demanded is a constant, minimizing waste is equivalent to minimizing total material area consumed:

minimize  sum_j (roll_width_j × x_j)

where

x_j

is the length cut using pattern

. The waste area is then:

waste = total_material_area − required_useful_area

where

required_useful_area = sum_i (order_width_i × order_length_i)

由于需求的总有用面积为常数，最小化废料等价于最小化消耗的总材料面积：

minimize  sum_j (roll_width_j × x_j)

其中

x_j

为使用模式

切割的长度。废料面积为：

waste = total_material_area − required_useful_area

其中

required_useful_area = sum_i (order_width_i × order_length_i)

。

Gotcha

注意事项

Using

sum_j (waste_width_j × x_j)

as the objective only captures trim loss — the unused strip within each pattern. It does not penalize over-production of an order. The solver will over-produce narrow orders to fill patterns efficiently, but that excess material is still waste. Always use total material area as the objective.

使用

sum_j (waste_width_j × x_j)

作为目标函数仅能捕捉切料损耗——每个模式内未使用的条料。它不会惩罚订单的过度生产。求解器会过度生产窄幅订单以高效填充模式，但这些多余材料仍属于废料。务必使用总材料面积作为目标函数。

Goal programming (preemptive / lexicographic)

目标规划（优先/字典序）

Goal programming optimizes multiple objectives in priority order. Implement it as sequential solves — one per priority level.

目标规划按优先级顺序优化多个目标。实现方式为顺序求解——每个优先级对应一次求解。

Formulation pattern

建模模式

Hard constraints — capacity limits, non-negativity, etc. These hold in every phase.
Goal constraints — for each goal, introduce deviation variables (d⁻ for underachievement, d⁺ for overachievement) and write an equality:
```
expression + d⁻ − d⁺ = target
```
.
Solve sequentially by priority:
- Phase 1: minimize (or maximize) the relevant deviation for the highest-priority goal.
- Phase k: fix all higher-priority deviations at their optimal values, then optimize priority k's deviation.

硬约束——产能限制、非负性等。这些约束在每个阶段均需满足。
目标约束——为每个目标引入偏差变量（d⁻表示未达成目标，d⁺表示超额达成目标），并写出等式：
```
表达式 + d⁻ − d⁺ = 目标值
```
。
按优先级顺序求解：
- 阶段1：最小化（或最大化）最高优先级目标的相关偏差。
- 阶段k：将所有更高优先级的偏差固定在其最优值，然后优化第k优先级目标的偏差。

Variable types in goal programming

目标规划中的变量类型

Deviation variables (d⁻, d⁺) and slack/idle-time variables are always continuous. However, decision variables must still be INTEGER when they represent discrete/countable quantities (units produced, vehicles, workers, etc.). Do not let the presence of continuous deviation variables cause you to make all variables continuous — the integrality of decision variables directly affects feasibility and objective values.

偏差变量（d⁻、d⁺）和松弛/闲置时间变量始终为连续变量。然而，决策变量若代表离散/可数数量（产量、车辆、工人等），仍必须为整数。不要因存在连续偏差变量而将所有变量设为连续——决策变量的整数性直接影响可行性和目标值。

Multi-period inventory / purchasing models

多周期库存/采购模型

In problems with buying, selling, and warehouse capacity over multiple periods, decide which capacity constraints to include based on the problem's timing assumptions.

在涉及多周期采购、销售和仓库容量的问题中，需根据问题的时间假设决定包含哪些容量约束。

Pattern

模式

For each period t with inventory balance

stock[t] = stock[t-1] + buy[t] - sell[t]

End-of-period capacity (variable bound):
```
stock[t] <= capacity
```
— always needed.
After-purchase capacity (explicit constraint):
```
stock[t-1] + buy[t] <= capacity
```
— prevents buying more than the warehouse can hold before any sales occur within the period.

对于每个周期t，库存余额为

stock[t] = stock[t-1] + buy[t] - sell[t]

：

期末容量（变量边界）：
```
stock[t] <= capacity
```
——始终需要。
采购后容量（显式约束）：
```
stock[t-1] + buy[t] <= capacity
```
——防止在周期内销售前采购超出仓库容量的货物。

When to include the after-purchase constraint

何时包含采购后约束

Include it when the problem states or implies that purchases are received before sales happen within a period (sequential operations), or when the warehouse physically cannot exceed capacity at any instant.
Omit it when buying and selling are concurrent within a period (common in textbook trading/inventory problems) and the capacity applies only to end-of-period stock. Many classic problems only constrain end-of-period inventory.

Key interaction with the sell constraint: If the model already has

sell[t] <= stock[t-1]

(grain bought this period cannot be sold this period), the model is bounded even without the after-purchase constraint. The sell constraint prevents unbounded buy-sell cycling. The after-purchase constraint is then an additional physical restriction, not a mathematical necessity.

Default: If the problem does not specify timing within a period, use only end-of-period capacity (

stock[t] <= capacity

). Add the after-purchase constraint only if the problem explicitly requires it.

包含：当问题说明或暗示周期内采购先于销售（顺序操作），或仓库任何时刻均不能超出容量时。
省略：当周期内采购和销售同时进行（常见于教材中的交易/库存问题）且容量仅适用于期末库存时。许多经典问题仅约束期末库存。

与销售约束的关键交互：若模型已包含

sell[t] <= stock[t-1]

（本期采购的货物不可在本期销售），即使没有采购后约束，模型也有界。销售约束可防止无限制的采购-销售循环。采购后约束是额外的物理限制，而非数学必需条件。

默认规则：若问题未指定周期内的时间顺序，仅使用期末容量（

stock[t] <= capacity

）。仅当问题明确要求时，才添加采购后约束。

Blending with shared mixing / intermediate processing

带共享混合/中间加工的配料问题

In some blending problems, a subset of raw materials must be mixed together first (e.g., in a mixing tank) before being allocated to different products. The resulting intermediate has a uniform composition — you cannot independently assign different raw materials to different products.

在某些配料问题中，部分原材料必须先混合（例如在混合罐中），然后再分配给不同产品。生成的中间产物具有均匀成分——无法独立将不同原材料分配给不同产品。

Why the standard blending LP is wrong here

标准配料LP为何在此处失效

The standard blending LP uses variables

x[i][j]

(amount of raw material

in product

) and freely allocates each raw material to each product. When raw materials share a mixing step, the proportions of those raw materials must be identical in every product that receives the intermediate. This proportionality constraint is bilinear (

x[A,1]*x[B,2] = x[B,1]*x[A,2]

) and cannot be directly expressed in an LP.

标准配料LP使用变量

x[i][j]

（原材料

在产品

中的用量），可自由将每种原材料分配给每种产品。当原材料共享混合步骤时，这些原材料的比例在所有接收中间产物的产品中必须完全相同。这种比例约束为双线性（

x[A,1]*x[B,2] = x[B,1]*x[A,2]

），无法直接在LP中表达。

Linearization strategies

线性化策略

Single-product allocation: If analysis shows the intermediate is profitable in only one product, allocate all intermediate to that product (set intermediate allocation to other products to zero). The proportionality constraint becomes trivially satisfied. This is the most common case — check profitability of intermediate in each product before attempting a general split.
Parametric over intermediate concentration: Fix the sulfur/quality concentration of the intermediate as a parameter
```
σ
```
. For each fixed
```
σ
```
, the problem is a standard LP (intermediate becomes a virtual raw material with known properties). Solve for a grid of
```
σ
```
values or use the structure to find the optimum analytically.
Scenario enumeration: When only 2–3 products exist, enumerate which products receive the intermediate (all-to-A, all-to-B, split). For each scenario with a single recipient, the LP is standard. For split scenarios, use strategy 2.

单产品分配：若分析表明中间产物仅在一种产品中盈利，将所有中间产物分配给该产品（设置中间产物向其他产品的分配量为0）。比例约束自动满足。这是最常见的情况——在尝试通用拆分前，先检查中间产物在每种产品中的盈利性。
基于中间产物浓度的参数化：将中间产物的硫含量/质量浓度固定为参数
```
σ
```
。对于每个固定的
```
σ
```
，问题变为标准LP（中间产物成为具有已知属性的虚拟原材料）。求解一系列
```
σ
```
值或利用结构分析找出最优解。
场景枚举：当仅存在2-3种产品时，枚举哪些产品接收中间产物（全部分配给A、全部分配给B、拆分分配）。对于单接收方的场景，LP为标准形式。对于拆分场景，使用策略2。

Profitability check

盈利性检查

Before formulating, check whether using the intermediate in each product is profitable:

Compare the minimum cost per ton of the intermediate (using cheapest feasible raw material mix) against each product's selling price.
If
```
cost_intermediate > sell_price[j]
```
for some product
```
j
```
, the intermediate should not be allocated to product
```
j
```
. Raw material C (or other direct inputs) alone may also be unprofitable if
```
cost_C > sell_price[j]
```
.
This analysis often eliminates the need for a bilinear split entirely.

建模前，检查中间产物在每种产品中的使用是否盈利：

比较中间产物的最低单位成本（使用最便宜的可行原材料组合）与每种产品的售价。
若
```
cost_intermediate > sell_price[j]
```
对于某产品
```
j
```
，中间产物不应分配给该产品。单独使用原材料C（或其他直接输入）若
```
cost_C > sell_price[j]
```
也可能不盈利。
这种分析通常可完全消除双线性拆分的需求。