optimization-ai999x — offline, verified optimization solver with step-by-step answers

Install · Capability guide · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.0

Field: Mathematics & Statistics · Tasks: Question Answering, Sentence Similarity, Feature Extraction

optimization-ai999x is an offline, deterministic Python solver that answers optimization questions with verified results from 72 certified capabilities, gives 240 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

optimization-ai999x is a compact, deterministic optimization engine for Python. It reads a question in plain English (or a typed call), picks a certified optimization capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Highlights

• Verified, not generated. Every answer is computed by a certified optimization capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 240 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: linear and integer programming, convex and nonlinear optimisation, assignment, transportation and shortest-route problems, scheduling, inventory, queueing, game theory and multi-criteria decisions.

Model overview

Property	Value
Package	optimization-ai999x 1.0.0 (import optimization_ai999x)
Type	Deterministic neuro-symbolic optimization engine (rule-based reading + certified symbolic capabilities)
Capabilities	72 certified procedures
Knowledge pack	version 1.0.1
Golden answers	240 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Quickstart

pip install optimization-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/optimization-ai999x/resolve/main/dist/optimization_ai999x-1.0.0-py3-none-any.whl

import optimization_ai999x as optimization

optimization.ask("Solve the transportation problem with supply [20, 30], demand [10, 25, 15] and cost matrix [[2, 3, 1], [5, 4, 8]].").show()
# -> SOLVED — min_cost = 150, shipments = [[5.0, 0.0, 15.0], [5.0, 25.0, 0.0]], plan_unique = yes, dual = [-3.0, 0.0, 5.0, 4.0, 4.0]  (transportation_problem)
optimization.ask("Find the shortest path from A to D in the undirected graph with edges A-B 4, A-C 2, B-C 1, B-D 5, C-D 8.").show()
# -> SOLVED — distance = 8, path = ['A', 'C', 'B', 'D'], path_unique = yes  (shortest_path)
optimization.solve("eoq", D=1200, S=50, H=3).show()
# -> SOLVED — Q = 200, total_cost = 600, orders_per_period = 6, cycle_time = 0.166667  (eoq)
optimization.ask("Find the AHP consistency ratio for the pairwise judgments [3, 5, 2].").show()
# -> SOLVED — CR = 0.003185, CI = 0.0018473, lambda_max = 3.00369, weights = [0.6483290138222366, 0.2296507940626371, 0.12202019211512621]  (ahp_consistency_ratio)
optimization.ask("Find the pure-strategy Nash equilibria of the bimatrix game with row payoffs [[3, 0], [5, 1]] and column payoffs [[3, 5], [0, 1]].").show()
# -> SOLVED — count = 1, equilibria = [[2.0, 2.0]]  (pure_nash_equilibria)
optimization.ask("Who created you?").message
# -> optimization-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

Command line: optimization-ai999x "<question>", optimization-ai999x --capabilities. The outputs above were produced by running this exact code on optimization-ai999x 1.0.0.

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Capabilities

72 certified procedures. A few of them:

• Procedures: 0/1 knapsack problem; 2 x 2 zero-sum game (saddle point or mixed-strategy formula); AHP consistency ratio; AHP priority vector (principal eigenvector); AHP weights by the geometric mean method; Absolute extrema of a polynomial on a closed interval; All-pairs shortest distances (Floyd-Warshall); Assignment problem (Hungarian method), maximum total value; Assignment problem (Hungarian method), minimum total cost; Bin packing (minimum number of bins); Break-even quantity; Chromatic number (minimum graph colouring); Closed cylinder of least surface area for a given volume; Critical path method (project duration and slack); Critical point of a two-variable quadratic and its classification; Discounted Markov decision process (optimal values and policy); Dynamic lot sizing (Wagner-Whitin); EOQ with planned backorders; Earliest due date rule (minimum maximum lateness); Economic order quantity (EOQ); Economic production quantity (EPQ); Equality-constrained quadratic program (Lagrange multipliers / KKT); Erlang B blocking probability (M/M/c/c loss system); Integer linear program (branch and bound) …

optimization_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 72 formulas

Show every formula, by topic

Calculus optimisation / geometry

Closed cylinder of least surface area for a given volume

$$r^{\ast }={\left(\frac{V}{2\pi }\right)}^{1\mathrm{/}3},h^{\ast }=2r^{\ast }r^*\; =\; \backslash left(\backslash frac\{V\}\{2\backslash pi\}\backslash right)^\{1/3\},\; \backslash quad\; h^*\; =\; 2r^*$$

Largest rectangle for a given length of fencing

$$A_{\max }=\frac{L^2}{16}\text{ (4 sides)},\frac{L^2}{8}\text{ (3 sides)}A\_\{\backslash max\}\; =\; \backslash frac\{L^2\}\{16\}\; \backslash text\{\; (4\; sides)\},\; \backslash quad\; \backslash frac\{L^2\}\{8\}\; \backslash text\{\; (3\; sides)\}$$

Open box of maximum volume from a rectangular sheet

$$x^{\ast }=\frac{(L+W)-\sqrt{L^2-LW+W^2}}{6}x^*\; =\; \backslash frac\{(L+W)\; -\; \backslash sqrt\{L^2\; -\; LW\; +\; W^2\}\}\{6\}$$

Calculus optimisation / one variable

Absolute extrema of a polynomial on a closed interval

Calculus optimisation / several variables

Critical point of a two-variable quadratic and its classification

$$D=f_{xx}{f}_{yy}-f_{xy}^2=4ac-b^{2}D\; =\; f\_\{xx\}\; f\_\{yy\}\; -\; f\_\{xy\}^2\; =\; 4ac\; -\; b^2$$

Cost-volume-profit analysis

Break-even quantity

$$Q_{BE}=\frac{F}{p-v}Q\_\{BE\}\; =\; \backslash frac\{F\}\{p\; -\; v\}$$

Dynamic programming / Markov decision processes

Discounted Markov decision process (optimal values and policy)

$$V^{\ast }(s)=\underset{a}{\max }(R(s,a)+\gamma \sum _{s^{\mathrm{\prime }}}P(s^{\mathrm{\prime }}\mid s,a)V^{\ast }(s^{\mathrm{\prime }}))V^*(s)\; =\; \backslash max\_a\; \backslash Big(\; R(s,a)\; +\; \backslash gamma\; \backslash sum\_\{s\text{'}\}\; P(s\text{'}\; \backslash mid\; s,a)\backslash ,\; V^*(s\text{'})\; \backslash Big)$$

Dynamic programming / inventory

Dynamic lot sizing (Wagner-Whitin)

$$C_t=\underset{1\le s\le t}{\min }(C_{s-1}+K+h\sum _{u=s}^t(u-s)d_u)C\_t\; =\; \backslash min\_\{1\; \backslash le\; s\; \backslash le\; t\}\; \backslash Big(\; C\_\{s-1\}\; +\; K\; +\; h\; \backslash sum\_\{u=s\}^\{t\}\; (u\; -\; s)\backslash ,\; d\_u\; \backslash Big)$$

Economic optimisation / monopoly pricing

Profit-maximising output with linear demand

$$q^{\ast }=\frac{a-c}{2b},p^{\ast }=\frac{a+c}{2}q^*\; =\; \backslash frac\{a\; -\; c\}\{2b\},\; \backslash quad\; p^*\; =\; \backslash frac\{a\; +\; c\}\{2\}$$

Economic optimisation / pricing

Revenue-maximising price with linear demand

$$p^{\ast }=\frac{\alpha }{2\beta },R^{\ast }=\frac{{\alpha }^2}{4\beta }p^*\; =\; \backslash frac\{\backslash alpha\}\{2\backslash beta\},\; \backslash quad\; R^*\; =\; \backslash frac\{\backslash alpha^2\}\{4\backslash beta\}$$

Game theory / non-zero-sum games

Mixed-strategy Nash equilibrium of a 2 x 2 game

$$q=\frac{a_{22}-a_{12}}{a_{11}-a_{12}-a_{21}+a_{22}},p=\frac{b_{22}-b_{21}}{b_{11}-b_{12}-b_{21}+b_{22}}q\; =\; \backslash frac\{a\_\{22\}\; -\; a\_\{12\}\}\{a\_\{11\}\; -\; a\_\{12\}\; -\; a\_\{21\}\; +\; a\_\{22\}\},\backslash quad\; p\; =\; \backslash frac\{b\_\{22\}\; -\; b\_\{21\}\}\{b\_\{11\}\; -\; b\_\{12\}\; -\; b\_\{21\}\; +\; b\_\{22\}\}$$

Pure-strategy Nash equilibria of a bimatrix game

$$a_{rs}=\underset{k}{\max }{a}_{ks},b_{rs}=\underset{k}{\max }{b}_{rk}a\_\{rs\}\; =\; \backslash max\_k\; a\_\{ks\},\backslash quad\; b\_\{rs\}\; =\; \backslash max\_k\; b\_\{rk\}$$

Game theory / zero-sum games

2 x 2 zero-sum game (saddle point or mixed-strategy formula)

$$v=\frac{ad-bc}{a+d-b-c},p=\frac{d-c}{a+d-b-c},q=\frac{d-b}{a+d-b-c}v\; =\; \backslash frac\{ad\; -\; bc\}\{a\; +\; d\; -\; b\; -\; c\},\backslash quad\; p\; =\; \backslash frac\{d\; -\; c\}\{a\; +\; d\; -\; b\; -\; c\},\backslash quad\; q\; =\; \backslash frac\{d\; -\; b\}\{a\; +\; d\; -\; b\; -\; c\}$$

Two-person zero-sum game (value and optimal mixed strategies)

$$v=\underset{x\in {\mathrm{\Delta }}_m}{\max }\underset{j}{\min }\sum _i{x}_i{a}_{ij}=\underset{y\in {\mathrm{\Delta }}_n}{\min }\underset{i}{\max }\sum _j{a}_{ij}{y}_{j}v\; =\; \backslash max\_\{x\; \backslash in\; \backslash Delta\_m\}\; \backslash min\_\{j\}\; \backslash sum\_i\; x\_i\; a\_\{ij\}\; =\; \backslash min\_\{y\; \backslash in\; \backslash Delta\_n\}\; \backslash max\_i\; \backslash sum\_j\; a\_\{ij\}\; y\_j$$

Integer programming

Integer linear program (branch and bound)

$$\max \mathrm{/}\min c^{\mathrm{\top }}x\text{s.t. }Ax(\le ,\ge ,=)b,x\in {\mathbb{Z}}_{\ge 0}^n\backslash max\; /\; \backslash min\; \backslash ;\; c^\backslash top\; x\; \backslash quad\; \backslash text\{s.t.\; \}\; A\; x\; \backslash ;(\backslash le,\backslash ge,=)\backslash ;\; b,\backslash ;\; x\; \backslash in\; \backslash mathbb\{Z\}\_\{\backslash ge\; 0\}^n$$

Integer programming / covering

Minimum vertex cover (and maximum independent set)

$$\tau (G)=\min \{\mathrm{\mid }C\mathrm{\mid }:C\cap \{u,v\}\mathrm{\ne }\mathrm{\varnothing }\mathrm{\forall }uv\in E\},\alpha (G)=\mathrm{\mid }V\mathrm{\mid }-\tau (G)\backslash tau(G)\; =\; \backslash min\backslash \{\backslash vert\; C\backslash vert\; :\; C\; \backslash cap\; \backslash \{u,\; v\backslash \}\; \backslash neq\; \backslash emptyset\backslash \; \backslash forall\; uv\; \backslash in\; E\backslash \},\backslash quad\; \backslash alpha(G)\; =\; \backslash vert\; V\backslash vert\; -\; \backslash tau(G)$$

Minimum-cost set cover

$$\min \sum _i{c}_i{x}_i\text{ s.t. }\sum _{i\ni e}{x}_i\ge 1\mathrm{\forall }e,x\in \{0,1\}\backslash min\; \backslash sum\_i\; c\_i\; x\_i\backslash \; \backslash text\{s.t.\}\backslash \; \backslash sum\_\{i\; \backslash ni\; e\}\; x\_i\; \backslash ge\; 1\backslash \; \backslash forall\; e,\backslash \; x\; \backslash in\; \backslash \{0,1\backslash \}$$

Integer programming / graph colouring

Chromatic number (minimum graph colouring)

$$\chi (G)=\min \{k:\mathrm{\exists }c:V\to \{\mathrm{1..}k\},c(u)\mathrm{\ne }c(v)\mathrm{\forall }uv\in E\}\backslash chi(G)\; =\; \backslash min\backslash \{k\; :\; \backslash exists\backslash ,\; c:\; V\; \backslash to\; \backslash \{1..k\backslash \},\backslash \; c(u)\; \backslash neq\; c(v)\backslash \; \backslash forall\; uv\; \backslash in\; E\backslash \}$$

Integer programming / knapsack

0/1 knapsack problem

$$\max \sum _i{v}_i{x}_i\text{ s.t. }\sum _i{w}_i{x}_i\le W,x_i\in \{0,1\}\backslash max\; \backslash sum\_i\; v\_i\; x\_i\; \backslash \; \backslash text\{s.t.\}\backslash \; \backslash sum\_i\; w\_i\; x\_i\; \backslash le\; W,\backslash \; x\_i\; \backslash in\; \backslash \{0,1\backslash \}$$

Unbounded knapsack problem

$$V(c)=\max (V(c-1),\underset{i:w_i\le c}{\max }V(c-w_i)+v_i)V(c)\; =\; \backslash max\backslash big(V(c-1),\backslash \; \backslash max\_\{i:\; w\_i\; \backslash le\; c\}\; V(c\; -\; w\_i)\; +\; v\_i\backslash big)$$

Integer programming / location

Uncapacitated facility location

$$\underset{S\mathrm{\ne }\mathrm{\varnothing }}{\min }\sum _{i\in S}{f}_i+\sum _j\underset{i\in S}{\min }{c}_{ij}\backslash min\_\{S\; \backslash neq\; \backslash emptyset\}\; \backslash sum\_\{i\; \backslash in\; S\}\; f\_i\; +\; \backslash sum\_j\; \backslash min\_\{i\; \backslash in\; S\}\; c\_\{ij\}$$

Integer programming / packing

Bin packing (minimum number of bins)

$$\min \mathrm{\#}\text{bins }\ge \lceil \sum _j{s}_j\mathrm{/}C\rceil \backslash min\; \backslash \#\backslash text\{bins\}\; \backslash \; \backslash ge\backslash \; \backslash left\backslash lceil\; \backslash sum\_j\; s\_j\; /\; C\; \backslash right\backslash rceil$$

Inventory management

EOQ with planned backorders

$$Q^{\ast }=\sqrt{\frac{2DS}{H}\cdot \frac{H+B}{B}},s^{\ast }=Q^{\ast }\frac{H}{H+B}Q^*\; =\; \backslash sqrt\{\backslash frac\{2DS\}\{H\}\backslash cdot\backslash frac\{H+B\}\{B\}\},\; \backslash quad\; s^*\; =\; Q^*\; \backslash frac\{H\}\{H+B\}$$

Economic order quantity (EOQ)

$$Q^{\ast }=\sqrt{\frac{2DS}{H}},T{C}^{\ast }=\sqrt{2DSH}Q^*\; =\; \backslash sqrt\{\backslash frac\{2\; D\; S\}\{H\}\},\; \backslash quad\; TC^*\; =\; \backslash sqrt\{2\; D\; S\; H\}$$

Economic production quantity (EPQ)

$$Q^{\ast }=\sqrt{\frac{2DS}{H(1-D\mathrm{/}P)}}Q^*\; =\; \backslash sqrt\{\backslash frac\{2DS\}\{H(1\; -\; D/P)\}\}$$

Order quantity with quantity discounts (all-units)

$$TC(Q)=D{p}_k+\frac{DS}{Q}+\frac{i{p}_{k}Q}{2}TC(Q)\; =\; D\; p\_k\; +\; \backslash frac\{D\; S\}\{Q\}\; +\; \backslash frac\{i\; p\_k\; Q\}\{2\}$$

Periodic-review order-up-to level

$$S=\bar{d}(R+L)+z_{SL}{\sigma }_d\sqrt{R+L}S\; =\; \backslash bar\; d\; (R\; +\; L)\; +\; z\_\{SL\}\backslash ,\backslash sigma\_d\backslash sqrt\{R\; +\; L\}$$

Reorder point with safety stock

$$ROP=\bar{d}L+z_{SL}{\sigma }_d\sqrt{L}ROP\; =\; \backslash bar\; d\; L\; +\; z\_\{SL\}\backslash ,\backslash sigma\_d\; \backslash sqrt\{L\}$$

Inventory management / single-period

Newsvendor order quantity (discrete demand)

$$Q^{\ast }=\min \{d:F(d)\ge \frac{p-c}{p-s}\}Q^*\; =\; \backslash min\backslash \{\; d\; :\; F(d)\; \backslash ge\; \backslash tfrac\{p\; -\; c\}\{p\; -\; s\}\; \backslash \}$$

Newsvendor order quantity (normal demand)

$$F(Q^{\ast })=\frac{p-c}{p-s},Q^{\ast }=\mu +z\sigma F(Q^*)\; =\; \backslash frac\{p\; -\; c\}\{p\; -\; s\},\; \backslash quad\; Q^*\; =\; \backslash mu\; +\; z\backslash ,\backslash sigma$$

Newsvendor order quantity (uniform demand)

$$Q^{\ast }=a+\frac{p-c}{p-s}(b-a)Q^*\; =\; a\; +\; \backslash frac\{p\; -\; c\}\{p\; -\; s\}(b\; -\; a)$$

Linear programming

Linear program (simplex method)

$$\max \mathrm{/}\min c^{\mathrm{\top }}x\text{s.t. }Ax(\le ,\ge ,=)b,x\ge 0\backslash max\; /\; \backslash min\; \backslash ;\; c^\backslash top\; x\; \backslash quad\; \backslash text\{s.t.\; \}\; A\; x\; \backslash ;(\backslash le,\; \backslash ge,\; =)\backslash ;\; b,\backslash ;\; x\; \backslash ge\; 0$$

Two-variable linear program (graphical corner-point method)

$$z^{\ast }=\underset{v\in \text{corners}}{\max }{c}^{\mathrm{\top }}vz^*\; =\; \backslash max\_\{v\; \backslash in\; \backslash text\{corners\}\}\; c^\backslash top\; v$$

Linear programming / sensitivity analysis

Shadow prices of a linear program

$$y^{\mathrm{\top }}=c_B^{\mathrm{\top }}{B}^{-1},\frac{\mathrm{\partial }{z}^{\ast }}{\mathrm{\partial }{b}_i}=y_{i}y^\backslash top\; =\; c\_B^\backslash top\; B^\{-1\},\; \backslash qquad\; \backslash frac\{\backslash partial\; z^*\}\{\backslash partial\; b\_i\}\; =\; y\_i$$

Multi-criteria decision analysis

TOPSIS ranking (closeness to the ideal solution)

C_i = \frac{D_i^-}{D_i^+ + D_i^-},\quad D_i^\pm = \Big\\vert w \odot r_i - v^\pm \Big\\vert _2

Weighted product model

$$P_i=\prod _j{n}_{ij}^{w_j}P\_i\; =\; \backslash prod\_j\; n\_\{ij\}^\{\backslash ,w\_j\}$$

Weighted sum model (simple additive weighting)

$$S_i=\sum _j{w}_j{n}_{ij},n_{ij}=\frac{x_{ij}}{\underset{i}{\max }{x}_{ij}}\text{ or }\frac{\underset{i}{\min }{x}_{ij}}{x_{ij}}S\_i\; =\; \backslash sum\_j\; w\_j\; n\_\{ij\},\backslash quad\; n\_\{ij\}\; =\; \backslash frac\{x\_\{ij\}\}\{\backslash max\_i\; x\_\{ij\}\}\backslash \; \backslash text\{or\}\backslash \; \backslash frac\{\backslash min\_i\; x\_\{ij\}\}\{x\_\{ij\}\}$$

Multi-criteria decision analysis / AHP

AHP consistency ratio

$$CI=\frac{{\lambda }_{\max }-n}{n-1},CR=\frac{CI}{R{I}_n}CI\; =\; \backslash frac\{\backslash lambda\_\{\backslash max\}\; -\; n\}\{n\; -\; 1\},\backslash quad\; CR\; =\; \backslash frac\{CI\}\{RI\_n\}$$

AHP priority vector (principal eigenvector)

$$Pw={\lambda }_{\max }w,\sum _i{w}_i=1P\; w\; =\; \backslash lambda\_\{\backslash max\}\; w,\backslash quad\; \backslash sum\_i\; w\_i\; =\; 1$$

AHP weights by the geometric mean method

$$w_i=\frac{(\prod _j{a}_{ij}{)}^{1\mathrm{/}n}}{\sum _k(\prod _j{a}_{kj}{)}^{1\mathrm{/}n}}w\_i\; =\; \backslash frac\{(\backslash prod\_j\; a\_\{ij\})^\{1/n\}\}\{\backslash sum\_k\; (\backslash prod\_j\; a\_\{kj\})^\{1/n\}\}$$

Network optimisation / assignment

Assignment problem (Hungarian method), maximum total value

$$\underset{\sigma }{\min }\sum _i{c}_{i\sigma (i)}=\max \{\sum _i{u}_i+\sum _j{v}_j:u_i+v_j\le c_{ij}\}\backslash min\_\{\backslash sigma\}\; \backslash sum\_i\; c\_\{i\backslash sigma(i)\}\; =\; \backslash max\; \backslash Big\backslash \{\; \backslash sum\_i\; u\_i\; +\; \backslash sum\_j\; v\_j\; :\; u\_i\; +\; v\_j\; \backslash le\; c\_\{ij\}\; \backslash Big\backslash \}$$

Assignment problem (Hungarian method), minimum total cost

$$\underset{\sigma }{\min }\sum _i{c}_{i\sigma (i)}=\max \{\sum _i{u}_i+\sum _j{v}_j:u_i+v_j\le c_{ij}\}\backslash min\_\{\backslash sigma\}\; \backslash sum\_i\; c\_\{i\backslash sigma(i)\}\; =\; \backslash max\; \backslash Big\backslash \{\; \backslash sum\_i\; u\_i\; +\; \backslash sum\_j\; v\_j\; :\; u\_i\; +\; v\_j\; \backslash le\; c\_\{ij\}\; \backslash Big\backslash \}$$

Network optimisation / flows

Maximum flow and minimum cut (Edmonds-Karp)

$$\max \mathrm{\mid }f\mathrm{\mid }=\underset{(S,\bar{S})}{\min }c(S,\bar{S})\backslash max\; \backslash vert\; f\backslash vert\; =\; \backslash min\_\{(S,\; \backslash bar\; S)\}\; c(S,\; \backslash bar\; S)$$

Minimum-cost flow (transshipment)

$$\min \sum c_{ij}{x}_{ij}\text{ s.t. }\sum _j{x}_{vj}-\sum _i{x}_{iv}=b_v,0\le x_{ij}\le u_{ij}\backslash min\; \backslash sum\; c\_\{ij\}\; x\_\{ij\}\backslash \; \backslash text\{s.t.\}\backslash \; \backslash sum\_j\; x\_\{vj\}\; -\; \backslash sum\_i\; x\_\{iv\}\; =\; b\_v,\backslash \; 0\; \backslash le\; x\_\{ij\}\; \backslash le\; u\_\{ij\}$$

Network optimisation / matching

Maximum bipartite matching

$$\nu (G)=\tau (G)\text{ for bipartite }G\backslash nu(G)\; =\; \backslash tau(G)\; \backslash text\{\; for\; bipartite\; \}\; G$$

Maximum-weight matching (general graph)

$$\underset{M\text{ matching}}{\max }\sum _{e\in M}{w}_e\backslash max\_\{M\; \backslash text\{\; matching\}\}\; \backslash sum\_\{e\; \backslash in\; M\}\; w\_e$$

Network optimisation / routing

Travelling salesman tour (exact, Held-Karp)

$$g(S,k)=\underset{m\in S\setminus \{k\}}{\min }(g(S\setminus \{k\},m)+d_{mk})g(S,\; k)\; =\; \backslash min\_\{m\; \backslash in\; S\; \backslash setminus\; \backslash \{k\backslash \}\}\; \backslash big(g(S\; \backslash setminus\; \backslash \{k\backslash \},\; m)\; +\; d\_\{mk\}\backslash big)$$

Network optimisation / shortest routes

All-pairs shortest distances (Floyd-Warshall)

$$d_{ij}\leftarrow \min (d_{ij},d_{ik}+d_{kj})d\_\{ij\}\; \backslash leftarrow\; \backslash min(d\_\{ij\},\backslash ,\; d\_\{ik\}\; +\; d\_\{kj\})$$

Shortest path (Dijkstra's algorithm)

$$d(v)=\underset{(u,v)}{\min }(d(u)+w_{uv})d(v)\; =\; \backslash min\_\{(u,v)\}\; \backslash big(d(u)\; +\; w\_\{uv\}\backslash big)$$

Shortest path with negative weights (Bellman-Ford)

$$d(v)\le d(u)+w_{uv}\mathrm{\forall }(u,v)d(v)\; \backslash le\; d(u)\; +\; w\_\{uv\}\; \backslash \; \backslash forall\; (u,v)$$

Network optimisation / spanning trees

Minimum spanning tree (Kruskal's algorithm)

$$\underset{T\text{ spanning tree}}{\min }\sum _{e\in T}{w}_e\backslash min\_\{T\; \backslash text\{\; spanning\; tree\}\}\; \backslash sum\_\{e\; \backslash in\; T\}\; w\_e$$

Network optimisation / transportation

Transportation problem (minimum shipping cost)

$$\min \sum _{ij}{c}_{ij}{x}_{ij}\text{ s.t. }\sum _j{x}_{ij}\le s_i,\sum _i{x}_{ij}=d_j,x\ge 0\backslash min\; \backslash sum\_\{ij\}\; c\_\{ij\}\; x\_\{ij\}\backslash \; \backslash text\{s.t.\}\backslash \; \backslash sum\_j\; x\_\{ij\}\; \backslash le\; s\_i,\backslash \; \backslash sum\_i\; x\_\{ij\}\; =\; d\_j,\backslash \; x\; \backslash ge\; 0$$

Nonlinear optimisation / descent methods

Steepest descent step with exact line search (quadratic)

$$\alpha =\frac{g^{\mathrm{\top }}g}{g^{\mathrm{\top }}Qg},x_1=x_0-\alpha g\backslash alpha\; =\; \backslash frac\{g^\backslash top\; g\}\{g^\backslash top\; Q\; g\},\backslash quad\; x\_1\; =\; x\_0\; -\; \backslash alpha\; g$$

Nonlinear optimisation / quadratic

Equality-constrained quadratic program (Lagrange multipliers / KKT)

Unconstrained quadratic minimisation

$$x^{\ast }=-Q^{-1}c,f^{\ast }=\frac{1}{2}{c}^{\mathrm{\top }}{x}^{\ast }x^*\; =\; -Q^\{-1\}\; c,\backslash quad\; f^*\; =\; \backslash tfrac12\; c^\backslash top\; x^*$$

Project scheduling

Critical path method (project duration and slack)

$$E{S}_k=\underset{j\to k}{\max }(E{S}_j+d_j),L{S}_k=\underset{k\to s}{\min }L{S}_s-d_k,{\text{slack}}_k=L{S}_k-E{S}_{k}ES\_k\; =\; \backslash max\_\{j\; \backslash to\; k\}(ES\_j\; +\; d\_j),\backslash quad\; LS\_k\; =\; \backslash min\_\{k\; \backslash to\; s\}\; LS\_s\; -\; d\_k,\backslash quad\; \backslash text\{slack\}\_k\; =\; LS\_k\; -\; ES\_k$$

PERT project duration and completion probability

$$t_e=\frac{a+4m+b}{6},{\sigma }^2={\left(\frac{b-a}{6}\right)}^2,P(T\le d)=\mathrm{\Phi }\text{\hspace{0.17em}⁣}\left(\frac{d-\mu }{\sigma }\right)t\_e\; =\; \backslash frac\{a\; +\; 4m\; +\; b\}\{6\},\backslash ;\; \backslash sigma^2\; =\; \backslash left(\backslash frac\{b\; -\; a\}\{6\}\backslash right)^2,\backslash ;\; P(T\; \backslash le\; d)\; =\; \backslash Phi\backslash !\backslash left(\backslash frac\{d\; -\; \backslash mu\}\{\backslash sigma\}\backslash right)$$

Queueing theory

M/G/1 queue (Pollaczek-Khinchine formula)

$$L_q=\frac{{\lambda }^2{\sigma }_S^2+{\rho }^2}{2(1-\rho )}L\_q\; =\; \backslash frac\{\backslash lambda^2\; \backslash sigma\_S^2\; +\; \backslash rho^2\}\{2(1-\backslash rho)\}$$

M/M/1 queue

$$L=\frac{\rho }{1-\rho },W=\frac{1}{\mu -\lambda },W_q=\frac{\rho }{\mu -\lambda }L\; =\; \backslash frac\{\backslash rho\}\{1-\backslash rho\},\backslash ;\; W\; =\; \backslash frac\{1\}\{\backslash mu\; -\; \backslash lambda\},\backslash ;\; W\_q\; =\; \backslash frac\{\backslash rho\}\{\backslash mu-\backslash lambda\}$$

M/M/c queue (Erlang C)

$$C(c,a)=\frac{B(c,a)}{1-\rho (1-B(c,a))},L_q=C\frac{\rho }{1-\rho }C(c,\; a)\; =\; \backslash frac\{B(c,a)\}\{1\; -\; \backslash rho(1\; -\; B(c,a))\},\; \backslash quad\; L\_q\; =\; C\backslash ,\backslash frac\{\backslash rho\}\{1-\backslash rho\}$$

M/M/c/K queue (finite capacity)

Machine-repair (finite-source) queue

$${\pi }_n\propto \frac{N!}{(N-n)!}\frac{(\lambda \mathrm{/}\mu {)}^n}{n!\text{ or }c!c^{n-c}}\backslash pi\_n\; \backslash propto\; \backslash frac\{N!\}\{(N-n)!\}\backslash ,\backslash frac\{(\backslash lambda/\backslash mu)^n\}\{n!\backslash \; \backslash text\{or\}\backslash \; c!\backslash ,c^\{n-c\}\}$$

Queueing theory / staffing

Minimum number of servers for a service target (Erlang C)

$$c^{\ast }=\min \{c>\lambda \mathrm{/}\mu :C(c,\lambda \mathrm{/}\mu )\le \alpha \}c^*\; =\; \backslash min\backslash \{c\; >\; \backslash lambda/\backslash mu\; :\; C(c,\; \backslash lambda/\backslash mu)\; \backslash le\; \backslash alpha\backslash \}$$

Queueing theory / teletraffic

Erlang B blocking probability (M/M/c/c loss system)

$$B(c,A)=\frac{A^c\mathrm{/}c!}{\sum _{k=0}^c{A}^k\mathrm{/}k!}B(c,\; A)\; =\; \backslash frac\{A^c\; /\; c!\}\{\backslash sum\_\{k=0\}^\{c\}\; A^k\; /\; k!\}$$

Revenue management

Littlewood's rule (two-fare protection level)

$$P(D_1>y^{\ast })=\frac{p_2}{p_1},y^{\ast }=\mu +z\sigma P(D\_1\; >\; y^*)\; =\; \backslash frac\{p\_2\}\{p\_1\},\; \backslash quad\; y^*\; =\; \backslash mu\; +\; z\backslash ,\backslash sigma$$

Scheduling / flow shop

Johnson's rule (two-machine flow shop)

$$\min C_{\max }\text{ for }F2vert{C}_{\max }\backslash min\; C\_\{\backslash max\}\; \backslash text\{\; for\; \}\; F2\; \backslash \backslash vert\; C\_\{\backslash max\}$$

Scheduling / parallel machines

Minimum makespan on identical parallel machines

$$\underset{\text{assignment}}{\min }\underset{k}{\max }\sum _{j\to k}{p}_j\backslash min\_\{\backslash text\{assignment\}\}\; \backslash max\_k\; \backslash sum\_\{j\; \backslash to\; k\}\; p\_j$$

Scheduling / single machine

Earliest due date rule (minimum maximum lateness)

$$\min L_{\max }=\min \underset{j}{\max }(C_j-d_j):d_{[1]}\le d_{[2]}\le \dots \backslash min\; L\_\{\backslash max\}\; =\; \backslash min\; \backslash max\_j\; (C\_j\; -\; d\_j):\backslash ;\; d\_\{[1]\}\; \backslash le\; d\_\{[2]\}\; \backslash le\; \backslash dots$$

Moore-Hodgson algorithm (minimum number of tardy jobs)

$$\min \sum _j{U}_j\backslash min\; \backslash sum\_j\; U\_j$$

Shortest processing time rule (minimum mean flow time)

$$\min \frac{1}{n}\sum _j{C}_j:p_{[1]}\le p_{[2]}\le \dots \backslash min\; \backslash tfrac\{1\}\{n\}\backslash sum\_j\; C\_j:\backslash ;\; p\_\{[1]\}\; \backslash le\; p\_\{[2]\}\; \backslash le\; \backslash dots$$

Weighted shortest processing time (Smith's rule)

$$\min \sum _j{w}_j{C}_j:\frac{p_{[1]}}{w_{[1]}}\le \frac{p_{[2]}}{w_{[2]}}\le \dots \backslash min\; \backslash sum\_j\; w\_j\; C\_j:\backslash ;\; \backslash frac\{p\_\{[1]\}\}\{w\_\{[1]\}\}\; \backslash le\; \backslash frac\{p\_\{[2]\}\}\{w\_\{[2]\}\}\; \backslash le\; \backslash dots$$

Unconstrained optimisation / one variable

Optimum of a quadratic function

$$x^{\ast }=-\frac{b}{2a},f^{\ast }=c-\frac{b^2}{4a}x^*\; =\; -\backslash frac\{b\}\{2a\},\; \backslash quad\; f^*\; =\; c\; -\; \backslash frac\{b^2\}\{4a\}$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified optimization capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (linear and integer programming, convex and nonlinear optimisation, assignment, transportation and shortest-route problems, scheduling, inventory, queueing, game theory and multi-criteria decisions); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{optimization_ai999x_1_0_0,
  title  = {optimization-ai999x: an offline, verified optimization solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.0},
  url    = {https://huggingface.co/AI999X/optimization-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/optimization-ai999x/discussions or visit https://ai999x.com.

Support the project

optimization-ai999x is free and open source. If it helps you, you can support further development on Buy Me a Coffee.

Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com