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Mathematical modeling
Turns engineering, economic, biological and social questions into variables, assumptions, solvable models, validated answers and an honest limits section.
Model formulation and assumptions
Mathematical modeling process
Turn a real question into variables, assumptions, a solvable model, a solution and a check against data.
1 reviewed map connectionsAssumptions and simplification
State which effects are ignored and why, so the model stays solvable while remaining defensible against the data.
2 reviewed map connectionsDecision and state variables
Name every symbol with its unit and range before writing equations, because undefined symbols make results unverifiable.
1 reviewed map connectionsDimensional analysis
Check that both sides of every equation share the same units, and derive scaling groups such as the Reynolds number.
2 reviewed map connectionsDiscretization
Replace continua by grids, time steps or samples so a numerical solver can work, and study the resulting truncation error.
1 reviewed map connectionsModeling report structure
Answer with assumptions, model, solution, validation and limitations, so a reviewer can retrace each choice.
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Optimization models
Linear programming model
Maximise or minimise a linear objective subject to linear inequalities, the workhorse of allocation problems.
2 reviewed map connectionsFeasible region and graphical method
Plot the constraints to see the feasible polygon; the optimum sits at a vertex unless the objective is flat on an edge.
1 reviewed map connectionsSimplex method
Move vertex to vertex along the feasible polytope, each step improving the objective until no improving edge remains.
2 reviewed map connectionsDuality and shadow price
Every linear program has a dual; the dual value of a binding constraint is the marginal worth of relaxing it.
1 reviewed map connectionsInteger programming
Force some variables to be whole numbers or binary, which models choosing, counting and sequencing decisions.
1 reviewed map connectionsBranch and bound
Solve relaxed linear problems, branch on fractional integers and prune nodes whose bound cannot beat the incumbent.
1 reviewed map connectionsNonlinear programming and KKT conditions
For smooth nonlinear models, optimality requires stationarity, primal and dual feasibility and complementary slackness.
1 reviewed map connectionsMulti-objective optimization and Pareto front
Trade off competing objectives; the Pareto front collects solutions that no longer can be improved on one goal without hurting another.
1 reviewed map connectionsShortest path models
Dijkstra solves non-negative networks; Bellman-Ford tolerates negative edges and underpins route pricing.
1 reviewed map connectionsNetwork flow models
Maximise flow through capacities; the min-cut theorem turns a bottleneck argument into a computable bound.
1 reviewed map connectionsMinimum spanning tree
Connect all nodes at least total cost with Kruskal or Prim; the base of routing and pipeline design.
1 reviewed map connectionsTransportation and assignment problems
Ship supplies to demands or match workers to jobs at least cost; special structures beat generic solvers.
1 reviewed map connectionsKnapsack problem
Choose a subset under a capacity limit; dynamic programming solves it pseudo-polynomially.
1 reviewed map connectionsTraveling salesman routing
Visit each site once at least tour length; heuristics plus bounds are the practical modelling answer.
1 reviewed map connectionsDynamic programming recursions
Split a sequential decision into stages and store subproblem values so each is computed once.
1 reviewed map connectionsQueueing and inventory models
Balance arrival and service rates for waiting times, and trade ordering cost against holding cost for order size.
1 reviewed map connectionsPopulation and growth models
Write per-capita birth and death rates into a differential equation; the logistic term caps growth by carrying capacity.
1 reviewed map connectionsCompartmental differential equation models
Split the system into compartments with flows between them; the SIR model is the standard epidemic example.
1 reviewed map connectionsDifference equation models
Describe discrete-step dynamics such as generations or quarters, and study stability through the multiplier magnitude.
1 reviewed map connectionsMarkov chain prediction
Advance a state distribution by a transition matrix; its stationary vector is the long-run share of states.
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Evaluation and decision models
Analytic hierarchy process
Build pairwise comparison matrices over criteria, then read weights from the dominant eigenvector and check consistency.
1 reviewed map connectionsTOPSIS ranking
Score each alternative by its relative closeness to an ideal point and distance from an anti-ideal point.
1 reviewed map connectionsEntropy weighting
Let the data set criterion weights from dispersion: a more uniform column carries less information and gets a smaller weight.
1 reviewed map connectionsFuzzy comprehensive evaluation
Map graded judgements through membership functions and a fuzzy relation to a single comparable score.
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Data and statistical models
Regression modeling
Fit a conditional mean function to data, then interpret coefficients only within the range the design covers.
1 reviewed map connectionsLeast squares and residual diagnostics
Minimise squared residuals, then read residual plots for curvature, heteroscedasticity and outliers before trusting R-squared.
1 reviewed map connectionsInterpolation versus curve fitting
Interpolation passes through every sample, while fitting smooths noise by trading bias against variance.
1 reviewed map connectionsStandardization and scaling
Rescale columns before distance-based methods, otherwise the largest unit dominates the result.
1 reviewed map connectionsOutlier handling
Distinguish data errors from genuine extremes; robust fits and winsorising change conclusions in different ways.
1 reviewed map connectionsPrincipal component analysis
Rotate correlated variables onto orthogonal directions of maximal variance and keep the leading components.
1 reviewed map connectionsCluster analysis
Group objects by a distance metric; the metric and the linkage choice decide the partition as much as the algorithm does.
2 reviewed map connectionsDiscriminant analysis
Find linear combinations that separate labelled groups and classify new records by the closest group centre.
1 reviewed map connectionsTime series and exponential smoothing
Model level, trend and seasonality over time; smoothing weights recent observations more heavily than old ones.
1 reviewed map connectionsARIMA and differencing
Difference until stationary, fit autoregressive and moving average terms, and forecast with error growing in the horizon.
1 reviewed map connectionsMonte Carlo simulation
Sample the uncertain inputs many times to obtain an output distribution and confidence intervals for the answer.
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Validation and sensitivity
Sensitivity and robustness analysis
Re-run the model with parameters perturbed to see which assumptions actually drive the conclusion.
2 reviewed map connectionsModel validation and error measures
Hold out data, compare units and magnitudes, and report errors that a reader can act on.
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Common modelling mistakes
Overfitting and leakage
A model tuned on the test set or fed future information looks excellent and fails on new data.
1 reviewed map connectionsUnit and scale inconsistency
Mixing kilometres with hours or percent with proportion is the most common silent modelling error.
1 reviewed map connectionsCopying templates without assumptions
Reusing a famous model without stating its preconditions hides why the answer stops being valid.
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