jsLPSolver ========== [A linear programming solver for the rest of us!](https://youtu.be/LbfMmCf5-ds?t=51) ## What Can I do with it? You can solve problems that fit the following fact pattern like this one from [this](http://math.stackexchange.com/questions/59429/berlin-airlift-linear-optimization-problem) site. >On June 24, 1948, the former Soviet Union blocked all land and water routes through East Germany to Berlin. >A gigantic airlift was organized using American and British planes to supply food, clothing and other supplies >to more than 2 million people in West Berlin. > >The cargo capacity was 30,000 cubic feet for an American plane and 20,000 cubic feet for a British plane. >To break the Soviet blockade, the Western Allies had to maximize cargo capacity, >but were subject to the following restrictions: No more than 44 planes could be used. The larger American planes required 16 >personnel per flight; double that of the requirement for the British planes. The total number of personnel >available could not exceed 512. The cost of an American flight was $9000 and the cost of a British flight was $5000. >The total weekly costs could note exceed $300,000. >Find the number of American and British planes that were used to maximize cargo capacity. So How Would I Do This? ----------------------- Part of the reason I built this library is that I wanted to do as little thinking / setup as possible to solve the actual problem. Instead of tinkering with arrays to solve this problem, you would create a model in a JavaScript object, and solve it through the object's `solve` function; like this: ```javascript var solver = require("./src/solver"), results, model = { "optimize": "capacity", "opType": "max", "constraints": { "plane": {"max": 44}, "person": {"max": 512}, "cost": {"max": 300000} }, "variables": { "brit": { "capacity": 20000, "plane": 1, "person": 8, "cost": 5000 }, "yank": { "capacity": 30000, "plane": 1, "person": 16, "cost": 9000 } }, }; results = solver.Solve(model); console.log(results); ``` which should yield the following: ``` {feasible: true, brit: 24, yank: 20, result: 1080000} ``` What If I Want Only Integers -------------------- Say you live in the real world and partial results aren't realistic, too messy, or generally unsafe. > You run a small custom furniture shop and make custom tables and dressers. > > Each week you're limited to 300 square feet of wood, 110 hours of labor, > and 400 square feet of storage. > > A table uses 30sf of wood, 5 hours of labor, requires 30sf of storage and has a > gross profit of $1,200. A dresser uses 20sf of wood, 10 hours of work to put > together, requires 50 square feet to store and has a gross profit of $1,600. > > How much of each do you produce to maximize profit, given that partial furniture > aren't allowed in this dumb world problem? ```javascript var solver = require("./src/solver"), model = { "optimize": "profit", "opType": "max", "constraints": { "wood": {"max": 300}, "labor": {"max": 110}, "storage": {"max": 400} }, "variables": { "table": {"wood": 30, "labor": 5, "profit": 1200, "table": 1, "storage": 30}, "dresser": {"wood": 20, "labor": 10, "profit": 1600, "dresser": 1, "storage": 50} }, "ints": {"table": 1, "dresser": 1} } console.log(solver.Solve(model)); // {feasible: true, result: 1440-0, table: 8, dresser: 3} ``` How Fast is it? ---------------------- Below are the results from my home made suite of variable sized LP(s) ```javascript { Relaxed: { variables: 2, time: 0.004899597 }, Unrestricted: { constraints: 1, variables: 2, time: 0.001273972 }, 'Chevalier 1': { constraints: 5, variables: 2, time: 0.000112002 }, Artificial: { constraints: 2, variables: 2, time: 0.000101994 }, 'Wiki 1': { variables: 3, time: 0.000358714 }, 'Degenerate Min': { constraints: 5, variables: 2, time: 0.000097377 }, 'Degenerate Max': { constraints: 5, variables: 2, time: 0.000085829 }, 'Coffee Problem': { constraints: 2, variables: 2, time: 0.000296747 }, 'Computer Problem': { constraints: 2, variables: 2, time: 0.000066585 }, 'Generic Business Problem': { constraints: 2, variables: 2, time: 0.000083135 }, 'Generic Business Problem 2': { constraints: 2, variables: 2, time: 0.000040413 }, 'Chocolate Problem': { constraints: 2, variables: 2, time: 0.000058503 }, 'Wood Shop Problem': { constraints: 2, variables: 2, time: 0.000045416 }, 'Integer Wood Problem': { constraints: 3, variables: 2, ints: 2, time: 0.002406691 }, 'Berlin Air Lift Problem': { constraints: 3, variables: 2, time: 0.000077362 }, 'Integer Berlin Air Lift Problem': { constraints: 3, variables: 2, ints: 2, time: 0.000823271 }, 'Infeasible Berlin Air Lift Problem': { constraints: 5, variables: 2, time: 0.000411828 }, 'Integer Wood Shop Problem': { constraints: 2, variables: 3, ints: 3, time: 0.001610363 }, 'Integer Sports Complex Problem': { constraints: 6, variables: 4, ints: 4, time: 0.001151579 }, 'Integer Chocolate Problem': { constraints: 2, variables: 2, ints: 2, time: 0.000109692 }, 'Integer Clothing Shop Problem': { constraints: 2, variables: 2, ints: 2, time: 0.000382191 }, 'Integer Clothing Shop Problem II': { constraints: 2, variables: 4, ints: 4, time: 0.000113927 }, 'Shift Work Problem': { constraints: 6, variables: 6, time: 0.000127012 }, 'Monster Problem': { constraints: 576, variables: 552, time: 0.054285454 }, monster_II: { constraints: 842, variables: 924, ints: 112, time: 0.649073406 } } ``` Alternative Model Formats ----------- Part of the reason that I build this library is because I *hate* setting up tableaus. Sometimes though, its just easier just to work with a tableau; other times, javaScript might not be the right environment to solve linear programs in. Fear not, we (kind of) have you covered! __USING A TABLEAU__ The tableau "style-guide" I used to parse LPs came from [LP_Solve](http://lpsolve.sourceforge.net/5.5/lp-format.htm). * Get LP From File * ```javascript var fs = require("fs"), solver = require("./src/solver"), model = {}; // Read Data from File fs.readFile("./your/model/file", "utf8", function(e,d){ // Convert the File Data to a JSON Model model = solver.ReformatLP(d); // Solve the LP console.log(solver.Solve(model)); }); ``` * Get LP From Arrays * ```javascript var solver = require("./src/solver"), model = [ "max: 1200 table 1600 dresser", "30 table 20 dresser <= 300", "5 table 10 dresser <= 110", "30 table 50 dresser <= 400", "int table", "int dresser", ]; // Reformat to JSON model model = solver.ReformatLP(model); // Solve the model solver.Solve(model); ``` __EXPORTING A TABLEAU__ You can also exports JSON models to an [LP_Solve](http://lpsolve.sourceforge.net/5.5/lp-format.htm) format (which is convenient if you plan on using LP_Solve). ```javascript var solver = require("./src/solver"), fs = require("fs"), model = { "optimize": "profit", "opType": "max", "constraints": { "wood": {"max": 300}, "labor": {"max": 110}, "storage": {"max": 400} }, "variables": { "table": {"wood": 30, "labor": 5, "profit": 1200, "table": 1, "storage": 30}, "dresser": {"wood": 20, "labor": 10, "profit": 1600, "dresser": 1, "storage": 50} }, "ints": {"table": 1, "dresser": 1} }; // convert the model to a string model = solver.ReformatLP(model); // Push the string to file fs.writeFile("./something.LP", model); ``` Multi-Objective Optimization ---------------------------- __What is it?__ Its a way to "solve" linear programs with multiple objective functions. Basically, it solves each objective function independent of one another and returns an object with: * The mid-point between those solutions (midpoint) * The solutions themselves (vertices) * The range of values for each variable in the solution (ranges) __Caveats__ I have no idea if this is right or not. Proceed with caution. __Use Case__ Say you're a dietitician and have a client that has the following constraints in their diet: * 375g of carbohydrates / day * 225g of protein / day * 66.66g of fat / day They're also a bit of a junk-food glutton and would like you to create a meal for them containing the following ingredients which meets their nutrition specifications outlined above: * egg whites * whole eggs * cheddar cheese * bacon * potato * fries They also mention they want to eat as much bacon, cheese, and fries as possible. After losing your appetite, you build the following linear program: ```javascript { "optimize": { "bacon": "max", "cheddar cheese": "max", "french fries": "max" }, "constraints": { "carb": { "equal": 375 }, "protein": { "equal": 225 }, "fat": { "equal": 66.666 } }, "variables": { "egg white":{ "carb": 0.0073, "protein": 0.109, "fat": 0.0017, "egg white": 1 }, "egg whole":{ "carb": 0.0072, "protein": 0.1256, "fat": 0.0951, "egg whole": 1 }, "cheddar cheese":{ "carb": 0.0128, "protein": 0.249, "fat": 0.3314, "cheddar cheese": 1 }, "bacon":{ "carb": 0.00667, "protein": 0.116, "fat": 0.4504, "bacon": 1 }, "potato": { "carb": 0.1747, "protein": 0.0202, "fat": 0.0009, "potato": 1 }, "french fries": { "carb": 0.3902, "protein": 0.038, "fat": 0.1612, "french fries": 1 } } } ``` which is solved by the ```solver.MultiObjective``` method. The result for this problem is: ```javascript { midpoint: { feasible: true, result: -0, 'egg white': 1494.64994046, potato: 1788.20687788, bacon: 46.02690209, 'cheddar cheese': 63.03985067, 'french fries': 129.61405521 }, vertices: [ { bacon: 138.08070627, 'egg white': 1532.31628515, potato: 2077.23579169, 'cheddar cheese': 0, 'french fries': 0 }, { 'cheddar cheese': 189.119552, 'egg white': 1246.61767465, potato: 2080.58935724, bacon: 0, 'french fries': 0 }, { 'french fries': 388.84216563, 'egg white': 1705.0158616, potato: 1206.79548473, bacon: 0, 'cheddar cheese': 0 } ], ranges: { bacon: { min: 0, max: 138.08070627 }, 'egg white': { min: 1246.61767465, max: 1705.0158616 }, potato: { min: 1206.79548473, max: 2080.58935724 }, 'cheddar cheese': { min: 0, max: 189.119552 }, 'french fries': { min: 0, max: 388.84216563 } } } ``` __Reading the Results__ midpoint: This is the solution that maximizes the amount of bacon, cheese, and fries your client can eat *simultaneously*. No single objective can be improved without hurting at least one other objective. vertices: The solutions to the individual objective functions ranges: This tells you the absolute minimum and absolute maximum values for each variable that was encountered while solving the objective functions. For instance, I can have between 0 and 138 grams of bacon. If I have 0 grams, I can have more cheese and more fries. If I have 138 grams, I can have no cheese and no fries.