{ "source_file": "0930", "source_platform": "Medium", "author": "Maxime Labonne", "title_listed": "Integer vs. Linear Programming in Python", "keywords": "整数规划,线性规划,优化", "domain": "机器学习", "content_type": "深度技术解析", "is_paid_flag": "部分免费/会员墙", "url": "https://towardsdatascience.com/integer-programming-vs-linear-programming-in-python-f1be5bb4e60e/", "publish_date": "2022-04", "corpus_stage": "预训练/SFT", "notes": "", "id": "db450d041200", "collected_at": "2026-10-01T22:13:43", "title": "Integer vs. Linear Programming in Python | Towards Data Science", "body_markdown": "![Revisit consent button](https://cdn-cookieyes.com/assets/images/revisit.svg)\nWe value your privacy\nWe use cookies to enhance your browsing experience, serve personalised ads or content, and analyse our traffic. 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It also helps in adverstising on the website.\n\n\n * Cookie\nedgebucket\n * Duration\nsession\n * Description\nReddit sets this cookie to save the information about a log-on Reddit user, for the purpose of advertisement recommendations and updating the content.\n\n\n * Cookie\ndid\n * Duration\n1 year\n * Description\nArbor sets this cookie to show targeted ads to site visitors.This cookie expires after 2 months or 1 year.\n\n\nUncategorised\nOther uncategorised cookies are those that are being analysed and have not been classified into a category as yet.\nNo cookies to display.\nReject All Save My Preferences Accept All\nToggle Search[](https://towardsdatascience.com/)Toggle Mobile Navigation\n[](https://towardsdatascience.com/)\nPublish AI, ML & data-science insights to a global community of data professionals.\nSign in[Submit an Article](https://write.insightmediagroup.io)\n[Latest](https://towardsdatascience.com/latest/)[Editor's Picks](https://towardsdatascience.com/tag/editors-pick/)[Newsletter](https://towardsdatascience.com/tag/the-variable/)[ShipAI](https://towardsdatascience.com/shipai/)\n* * *\n[Write For TDS](https://towardsdatascience.com/submissions/)\n[LinkedIn](https://www.linkedin.com/company/towards-data-science/)[X](https://x.com/TDataScience)Toggle Search\n[Artificial Intelligence](https://towardsdatascience.com/category/artificial-intelligence/)\n# Integer vs. Linear Programming in Python\nA guide to identify and solve any optimization problem\n[Maxime Labonne](https://towardsdatascience.com/author/mlabonne/)\nApril 7, 202213 min read\n### Mixed Integer Programming for optimization with Google OR-Tools\n![Image by author, emojis by OpenMoji \\(CC BY-SA 4.0\\)](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/161w8yJ9mSDplUK30hXhpQg.jpeg)\nImage by author, emojis by [OpenMoji](https://openmoji.org/) ([CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/))\nWhy is **linear programming** called that way?\nBoth terms are confusing:\n * **Linear** implies that **nonlinear** programming exists;\n * **Programming** actually **** means** \"plannin**g\" in this context.\n\n\nIn summary, it has nothing to do with code: linear or not. It's about **optimizing** variables with various constraints.\nIn this article, we're gonna talk about another type of optimization: **integer programming**. We'll see why a good understanding of the problem we face is necessary to choose the right solver. Finally, we will write a model that can take on a bigger challenge and actually solve a whole class of optimization problems.\nLearn this step by step with the interactive [AI Engineer](https://roadmap.sh/ai-engineer) and [Computer Science](https://roadmap.sh/computer-science) roadmaps.\nYou can run the code from this tutorial with the following [**Google Colab notebook**](https://colab.research.google.com/drive/1aIKnXPEiXFo1M3U0EM6arE2teD9KkJdF?usp=sharing).\n![Image by author, emojis by OpenMoji \\(CC BY-SA 4.0\\)](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/0Nh7gzKjRzcxFQo_o.png)\nImage by author, emojis by [OpenMoji](https://openmoji.org/) ([CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/))\n## 📊 I. Optimization problem types\nIn the [introduction to linear programming](https://towardsdatascience.com/introduction-to-linear-programming-in-python-9261e7eb44b), we **optimized an army composition**. Here was the result:\ntext\n\n```\n================= Solution =================\nSolved in 87.00 milliseconds in 2 iterations\n\nOptimal power = 1800.0 💪 power\nArmy:\n - 🗡 ️Swordsmen = 6.0000000000000036\n - 🏹 Bowmen = 0.0\n - 🐎 Horsemen = 5.999999999999999\n```\n\nHow can we have 5.999... horsemen? We specified that our variables **should be integers** with `VarInt`. What was wrong with our code?\nThe problem is not the model but the choice of the solver.\nGLOP is a [pure linear programming solver](https://en.wikipedia.org/wiki/GLOP). This means that it **cannot understand the concept of integers**. It is limited to continuous parameters with a linear relationship.\nThis is the difference between **linear** programming (LP) and **integer linear** programming (ILP). In summary, LP solvers can only use real numbers and not integers as variables. So why did we declare our variables as integers if it doesn't take them into account?\nGLOP cannot solve ILP problems, but other solvers can. Actually, a lot of them are **mixed integer linear programming** (MILP, commonly called MIP) solvers. This means that they can consider both **continuous** (real numbers) and **discrete** (integers) variables. A particular case of discrete values is Boolean variables to represent decisions with 0–1 values.\nOther solvers like SCIP or [CBC](https://projects.coin-or.org/Cbc) can solve both **MILP and MINLP** (mixed integer _nonlinear_ programming) problems. Thanks to OR-Tools, we can use the same model and just change the solver to SCIP or CBC.\npython\n\n```\n# Create the linear solver using the CBC backend\nsolver = pywraplp.Solver('Maximize army power', pywraplp.Solver.CBC_MIXED_INTEGER_PROGRAMMING)\n\n# 1. Create the variables we want to optimize\nswordsmen = solver.IntVar(0, solver.infinity(), 'swordsmen')\nbowmen = solver.IntVar(0, solver.infinity(), 'bowmen')\nhorsemen = solver.IntVar(0, solver.infinity(), 'horsemen')\n\n# 2. Add constraints for each resource\nsolver.Add(swordsmen*60 + bowmen*80 + horsemen*140 <= 1200)\nsolver.Add(swordsmen*20 + bowmen*10 <= 800)\nsolver.Add(bowmen*40 + horsemen*100 <= 600)\n\n# 3. Maximize the objective function\nsolver.Maximize(swordsmen*70 + bowmen*95 + horsemen*230)\n\n# Solve problem\nstatus = solver.Solve()\n\n# If an optimal solution has been found, print results\nif status == pywraplp.Solver.OPTIMAL:\n print('================= Solution =================')\n print(f'Solved in {solver.wall_time():.2f} milliseconds in {solver.iterations()} iterations')\n print()\n print(f'Optimal value = {solver.Objective().Value()} 💪 power')\n print('Army:')\n print(f' - 🗡 ️Swordsmen = {swordsmen.solution_value()}')\n print(f' - 🏹 Bowmen = {bowmen.solution_value()}')\n print(f' - 🐎 Horsemen = {horsemen.solution_value()}')\nelse:\n print('The solver could not find an optimal solution.')\n```\n\ntext\n\n```\n================= Solution =================\nSolved in 3.00 milliseconds in 0 iterations\n```\n\ntext\n\n```\nOptimal value = 1800.0 💪 power\nArmy: \n - 🗡 ️Swordsmen = 6.0\n - 🏹 Bowmen = 0.0\n - 🐎 Horsemen = 6.0\n```\n\nStrictly speaking, our variables are still floats (`type(swordsmen.solution_value()) = float`) but we can see that they don't have weird decimals anymore: the CBC solver really considered them as **integers**.\nIn this example, we would generally just **round up these values** since the error is insignificant. However, it is important to remember to choose the appropriate solver according to the studied problem:\n * **LP** for continuous variables;\n * **MIP/MILP** for a combination of continuous and discrete variables.\n\n\nThere are other types such as **quadratic** (QP) or **nonlinear** (NLP or MINLP, with an exponential objective function or constraints for instance) problems. They're applied in different contexts, but follow the same principles as LP or MIP solvers.\n![Image by author](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1eXPkPpMEX0bNxeRZnuXzKg.png)\nImage by author\n## 🧱 II. Building a general model\nBut what if our **resources change**? Or if the cost of a unit evolved? What if we upgraded horsemen and their power increased?\nOne of the best perks of OR-Tools is that it uses a general-purpose programming language like Python. Instead of static numbers, we can store our parameters in objects like **dictionaries** or **lists**.\nThe code won't be as readable, but it becomes much more flexible: actually, it can be so flexible that we can solve an **entire class of optimization problems** without changing the model (just the parameters).\nLet's transform our input parameters into Python lists and feed them to the solver through a function.\npython\n\n```\n# Inputs\nUNITS = ['🗡 ️Swordsmen', '🏹 Bowmen', '🐎 Horsemen']\n\nDATA = [[60, 20, 0, 70],\n [80, 10, 40, 95],\n [140, 0, 100, 230]]\n\nRESOURCES = [1200, 800, 600]\n\ndef solve_army(UNITS, DATA, RESOURCES):\n # Create the linear solver using the CBC backend\n solver = pywraplp.Solver('Maximize army power', pywraplp.Solver.CBC_MIXED_INTEGER_PROGRAMMING)\n\n # 1. Create the variables we want to optimize\n units = [solver.IntVar(0, solver.infinity(), unit) for unit in UNITS]\n\n # 2. Add constraints for each resource\n for r, _ in enumerate(RESOURCES):\n solver.Add(sum(DATA[u][r] * units[u] for u, _ in enumerate(units)) <= RESOURCES[r])\n\n # 3. Maximize the objective function\n solver.Maximize(sum(DATA[u][-1] * units[u] for u, _ in enumerate(units)))\n\n # Solve problem\n status = solver.Solve()\n\n # If an optimal solution has been found, print results\n if status == pywraplp.Solver.OPTIMAL:\n print('================= Solution =================')\n print(f'Solved in {solver.wall_time():.2f} milliseconds in {solver.iterations()} iterations')\n print()\n print(f'Optimal value = {solver.Objective().Value()} 💪 power')\n print('Army:')\n for u, _ in enumerate(units):\n print(f' - {units[u].name()} = {units[u].solution_value()}')\n else:\n print('The solver could not find an optimal solution.')\n\nsolve_army(UNITS, DATA, RESOURCES)\n```\n\ntext\n\n```\n================= Solution =================\nSolved in 2.00 milliseconds in 0 iterations\n```\n\ntext\n\n```\nOptimal value = 1800.0 💪 power \nArmy:\n - 🗡 ️Swordsmen = 6.0\n - 🏹 Bowmen = 0.0\n - 🐎 Horsemen = 6.0\n```\n\nWe obtain the same results: our code seems to work. Now let's **change the parameters** to tackle a slightly more complex problem.\nImagine we have a lot more resources: 🌾 **183000** , 🪵9**0512,** and 🪙80**150,** so we can also produce a lot more units! This is the new table:\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1NXG4ze1sme2tqOYLjW6Jyw.png)\nNotice that we transformed the 💪 **powe** r into two values: 💪 **atta** ck and ❤**️healt** h, which is a little more detailed. Health values are higher than attack values, which is why we want to add a weighting factor to make them more comparable.\nLet's take 10 as an example, so _power = 10*attack + health_. Our objective function becomes:\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1WGMoYgtG7I7PwJxLAtwSXg.png)\nAdapting our code to this new problem is actually quite simple: we just have to **change the input parameters** and update the **objective function**.\npython\n\n```\nUNITS = [\n '🗡 ️Swordsmen',\n '🛡 ️Men-at-arms',\n '🏹 Bowmen',\n '❌Crossbowmen',\n '🔫 Handcannoneers',\n '🐎 Horsemen',\n '♞Knights',\n '🐏 Battering rams',\n '🎯 Springalds',\n '🪨Mangonels',\n]\n\nDATA = [\n [60, 20, 0, 6, 70],\n [100, 0, 20, 12, 155],\n [30, 50, 0, 5, 70],\n [80, 0, 40, 12, 80],\n [120, 0, 120, 35, 150],\n [100, 20, 0, 9, 125],\n [140, 0, 100, 24, 230],\n [0, 300, 0, 200, 700],\n [0, 250, 250, 30, 200],\n [0, 400, 200, 12*3, 240]\n]\n\nRESOURCES = [183000, 90512, 80150]\n\ndef solve_army(UNITS, DATA, RESOURCES):\n # Create the linear solver using the CBC backend\n solver = pywraplp.Solver('Maximize army power', pywraplp.Solver.CBC_MIXED_INTEGER_PROGRAMMING)\n\n # 1. Create the variables we want to optimize\n units = [solver.IntVar(0, solver.infinity(), unit) for unit in UNITS]\n\n # 2. Add constraints for each resource\n for r, _ in enumerate(RESOURCES):\n solver.Add(sum(DATA[u][r] * units[u] for u, _ in enumerate(units)) <= RESOURCES[r])\n\n # 3. Maximize the new objective function\n solver.Maximize(sum((10*DATA[u][-2] + DATA[u][-1]) * units[u] for u, _ in enumerate(units)))\n\n # Solve problem\n status = solver.Solve()\n\n # If an optimal solution has been found, print results\n if status == pywraplp.Solver.OPTIMAL:\n print('================= Solution =================')\n print(f'Solved in {solver.wall_time():.2f} milliseconds in {solver.iterations()} iterations')\n print()\n print(f'Optimal value = {solver.Objective().Value()} 💪 power')\n print('Army:')\n for u, _ in enumerate(units):\n print(f' - {units[u].name()} = {units[u].solution_value()}')\n else:\n print('The solver could not find an optimal solution.')\n\nsolve_army(UNITS, DATA, RESOURCES)\n```\n\ntext\n\n```\n================= Solution =================\nSolved in 74.00 milliseconds in 412 iterations\n```\n\ntext\n\n```\nOptimal value = 1393145.0 💪 power\nArmy:\n - 🗡 ️Swordsmen = 2.0\n - 🛡 ️Men-at-arms = 1283.0\n - 🏹 Bowmen = 3.0\n - ❌Crossbowmen = 0.0\n - 🔫 Handcannoneers = 454.0\n - 🐎 Horsemen = 0.0\n - ♞Knights = 0.0\n - 🐏 Battering rams = 301.0\n - 🎯 Springalds = 0.0\n - 🪨Mangonels = 0.0\n```\n\nThis problem would take a long time for humans to address, but the ILP solver did it in the blink of an eye. Better than that: it also gives us the guarantee that **our solution is optimal** , which means that our enemy cannot find a better army composition for the same cost!\nWe could increase the number of units and give billions of resources but you get the picture: it would just take longer to obtain a solution, but it wouldn't change the problem.\n## ⚔️ III. Combining constraints\nNow, let's say we scouted our enemy and know that their army has a 💪 power of **1,000,00** 0. We could build a much better army, but our resources are precious and it wouldn't be very efficient: all we have to do is to build an army with a **💪 power higher than 1,000,0** 00 (even 1,000,001 would be enough).\nIn other words, the total power is now a **constraint** (💪 > 1,000,000) instead of the objective to maximize. The new goal is to minimize the resources we need to produce this army. However, we can reuse our input parameters since they didn't change.\nThe new constraint can be translated as \"the sum of the power of the selected units must be strictly greater than 1,000,000\".\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1LAq7zYjqG_-njYqOlcdnQQ.png)\nIn code, we can loop through our units and resources to design this constraint.\nThe objective function also has to change. Our goal is to **minimize the sum of resources spent** to build the army.\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1a9PmrsH5Zl2nNWN3LaGG-A.png)\nOnce again, we can loop through our resources to implement it in OR-Tools.\npython\n\n```\ndef solve_army(UNITS, DATA, RESOURCES):\n # Create the linear solver using the CBC backend\n solver = pywraplp.Solver('Minimize resource consumption', pywraplp.Solver.CBC_MIXED_INTEGER_PROGRAMMING)\n\n # 1. Create the variables we want to optimize\n units = [solver.IntVar(0, solver.infinity(), unit) for unit in UNITS]\n\n # 2. Add power constraint\n solver.Add(sum((10 * DATA[u][-2] + DATA[u][-1]) * units[u] for u, _ in enumerate(units)) >= 1000001)\n\n # 3. Minimize the objective function\n solver.Minimize(sum((DATA[u][0] + DATA[u][1] + DATA[u][2]) * units[u] for u, _ in enumerate(units)))\n\n # Solve problem\n status = solver.Solve()\n\n # If an optimal solution has been found, print results\n if status == pywraplp.Solver.OPTIMAL:\n print('================= Solution =================')\n print(f'Solved in {solver.wall_time():.2f} milliseconds in {solver.iterations()} iterations')\n print()\n\n power = sum((10 * DATA[u][-2] + DATA[u][-1]) * units[u].solution_value() for u, _ in enumerate(units))\n print(f'Optimal value = {solver.Objective().Value()} 🌾 🪵🪙resources')\n print(f'Power = 💪 {power}')\n print('Army:')\n for u, _ in enumerate(units):\n print(f' - {units[u].name()} = {units[u].solution_value()}')\n print()\n\n food = sum((DATA[u][0]) * units[u].solution_value() for u, _ in enumerate(units))\n wood = sum((DATA[u][1]) * units[u].solution_value() for u, _ in enumerate(units))\n gold = sum((DATA[u][2]) * units[u].solution_value() for u, _ in enumerate(units))\n print('Resources:')\n print(f' - 🌾 Food = {food}')\n print(f' - 🪵Wood = {wood}')\n print(f' - 🪙Gold = {gold}')\n else:\n print('The solver could not find an optimal solution.')\n\nsolve_army(UNITS, DATA, RESOURCES)\n```\n\ntext\n\n```\n================= Solution =================\nSolved in 4.00 milliseconds in 0 iterations\n```\n\ntext\n\n```\nOptimal value = 111300.0 🌾 🪵🪙resources\nPower = 💪 1001700.0 \nArmy:\n - 🗡 ️Swordsmen = 0.0\n - 🛡 ️Men-at-arms = 0.0\n - 🏹 Bowmen = 0.0\n - ❌Crossbowmen = 0.0\n - 🔫 Handcannoneers = 0.0\n - 🐎 Horsemen = 0.0\n - ♞Knights = 0.0\n - 🐏 Battering rams = 371.0\n - 🎯 Springalds = 0.0\n - 🪨Mangonels = 0.0\n```\n\ntext\n\n```\nResources:\n - 🌾 Food = 0.0\n - 🪵Wood = 111300.0\n - 🪙Gold = 0.0\n```\n\nThe solver found an optimal solution: we need to build **371 🐏 battering ram** s for a total cost of 111,300 🪵wood. Wait, what if we don't have that much wood? In the previous section, we only had 🪵90512: we cannot produce 371 🐏 battering rams. 😱\nSo is it possible to take these **limited resources** into account and still try to **build the best army**? Actually, it's super easy: we just have to copy/paste the constraints from the previous section.\nIn this version, we have two types of constraints:\n * The total power must be **greater than 1,000,000** ;\n * We cannot spend more than our **limited resources**.\n\n\npython\n\n```\ndef solve_army(UNITS, DATA, RESOURCES):\n # Create the linear solver using the CBC backend\n solver = pywraplp.Solver('Minimize resource consumption', pywraplp.Solver.CBC_MIXED_INTEGER_PROGRAMMING)\n\n # 1. Create the variables we want to optimize\n units = [solver.IntVar(0, solver.infinity(), unit) for unit in UNITS]\n\n # 2. Add constraints for each resource\n for r, _ in enumerate(RESOURCES):\n solver.Add(sum((10 * DATA[u][-2] + DATA[u][-1]) * units[u] for u, _ in enumerate(units)) >= 1000001)\n\n # Old constraints for limited resources\n for r, _ in enumerate(RESOURCES):\n solver.Add(sum(DATA[u][r] * units[u] for u, _ in enumerate(units)) <= RESOURCES[r])\n\n # 3. Minimize the objective function\n solver.Minimize(sum((DATA[u][0] + DATA[u][1] + DATA[u][2]) * units[u] for u, _ in enumerate(units)))\n\n # Solve problem\n status = solver.Solve()\n\n # If an optimal solution has been found, print results\n if status == pywraplp.Solver.OPTIMAL:\n print('================= Solution =================')\n print(f'Solved in {solver.wall_time():.2f} milliseconds in {solver.iterations()} iterations')\n print()\n\n power = sum((10 * DATA[u][-2] + DATA[u][-1]) * units[u].solution_value() for u, _ in enumerate(units))\n print(f'Optimal value = {solver.Objective().Value()} 🌾 🪵🪙resources')\n print(f'Power = 💪 {power}')\n print('Army:')\n for u, _ in enumerate(units):\n print(f' - {units[u].name()} = {units[u].solution_value()}')\n print()\n\n food = sum((DATA[u][0]) * units[u].solution_value() for u, _ in enumerate(units))\n wood = sum((DATA[u][1]) * units[u].solution_value() for u, _ in enumerate(units))\n gold = sum((DATA[u][2]) * units[u].solution_value() for u, _ in enumerate(units))\n print('Resources:')\n print(f' - 🌾 Food = {food}')\n print(f' - 🪵Wood = {wood}')\n print(f' - 🪙Gold = {gold}')\n else:\n print('The solver could not find an optimal solution.')\n\nsolve_army(UNITS, DATA, RESOURCES)\n```\n\ntext\n\n```\n================= Solution =================\nSolved in 28.00 milliseconds in 1 iterations\n```\n\ntext\n\n```\nOptimal value = 172100.0 🌾 🪵🪙resources\nPower = 💪 1000105.0\nArmy:\n - 🗡 ️Swordsmen = 1.0\n - 🛡 ️Men-at-arms = 681.0\n - 🏹 Bowmen = 0.0\n - ❌Crossbowmen = 0.0\n - 🔫 Handcannoneers = 0.0\n - 🐎 Horsemen = 0.0\n - ♞Knights = 0.0\n - 🐏 Battering rams = 301.0\n - 🎯 Springalds = 0.0\n - 🪨Mangonels = 0.0 \n```\n\ntext\n\n```\nResources:\n - 🌾 Food = 68160.0\n - 🪵Wood = 90320.0\n - 🪙Gold = 13620.0\n```\n\nSince we now have a **limited resource of 🪵wood,** the number of 🐏 battering rams sadly dropped from 371 to 301. In exchange, we got 681 🛡 ️men-at-arms and 1 lost 🗡 ️swordsman (welcome to them).\nThe total cost of the army is **172,100** , which is much higher than the 111,300 we previously found (+65% increase) but it truly is the optimal solution under these constraints. It shows that we should produce more wood because these 🐏 battering rams are extremely cost-efficient!\nThis example shows **how modular** LP models can be. It is possible to reuse parts of the code, like constraints, in another model to combine them and solve more complex problems.\n## 🧠 IV. Linear Programming vs Machine Learning\nLet's talk about the elephant in the room. Why not use **machine learning** (in a broad sense) instead of linear programming? It's not like this problem cannot be solved with a genetic algorithm for instance.\nMathematical optimization is often neglected in favor of machine learning techniques, but both have their merits:\n * Linear programming can produce an **optimal solution** in an undetermined amount of time (it can take years), while machine learning can approximate complex functions in no time.\n * There is **no training** in LP, but an expert is required to build a mathematical model. Machine learning needs data, but the models can be used as black boxes to solve a problem.\n * As a rule of thumb, problems that **do not have a particular time constraint** and/or are not extremely complex can be advantageously solved with linear programming.\n\n\n![Image by author, emojis by OpenMoji \\(CC BY-SA 4.0\\)](https://assets.insightmediagroup.io/media/wp-content/uploads/2022/04/1CkyNIGyMSguOt1eS9Uh10g.png)\nImage by author, emojis by [OpenMoji](https://openmoji.org/) ([CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/))\n## Conclusion\nIn this tutorial, we dived deeper into our understanding of mathematical optimization.\n * We talked about solvers and types of optimization problems: **LP, MIP, NLP** ;\n * We modeled and solved an extremely common optimization problem in an optimal way and **generalized our model** through a function;\n * We reframed this problem and **merged two sets of constraints** to obtain the best army composition for the lowest price;\n * We compared the **pros and cons** of linear programming and machine learning.\n\n\nThere are **a lot more problems** where optimization can be applied. For instance, how to create school timetables that satisfy everybody's requirements? How to deliver 1,000 different orders in a minimum amount of time? Where to create a new metro line to maximize its usefulness?\nIn future articles, we'll talk about new types of applications for these techniques, including satisfiability and nonlinear problems.\nI hope you enjoyed this more advanced article. If you like machine learning and optimization, [**let's connect on Twitter**](https://twitter.com/maximelabonne)!\n## Related articles\n> [**Chapter 3: Constraint Programming in Python**](https://towardsdatascience.com/constraint-programming-67ac16fa0c81)\n> [**Part 1: Introduction to Linear Programming in Python**](https://towardsdatascience.com/introduction-to-linear-programming-in-python-9261e7eb44b)\n* * *\n_Learn more about machine learning and support my work with one click - become a Medium member here:_\n> [**Join Medium with my referral link - Maxime Labonne**](https://medium.com/@mlabonne/membership)\n_If you're already a member, you can_[ _follow me on Medium_](https://medium.com/@mlabonne) _._\n## Register Now to Continue Reading\n### Unlock Towards Data Science with a Free Account\nAll articles on Towards Data Science are available to registered users. 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