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Linear programming applies mathematical equations to tackle business decisions. For instance, a retailer planning its Christmas inventory across four product lines can let linear programming compute the optimal production mix that maximizes profit.
Implementing linear programming requires translating the real-world problem into a mathematical model. The model defines an objective—commonly maximizing profit or minimizing cost—alongside decision variables and constraints that capture resources or limits. For example, a manufacturer with scarce raw materials must decide whether to focus on premium items or a larger volume of low‑cost goods; the model incorporates the objective, variables, and constraints to guide that decision.
Linear programming relies on linear equations: if you double sales while everything else stays constant, the equation shows a proportional increase in revenue. However, some decision variables exhibit non‑linear behavior. Doubling the budget for a startup, for instance, does not necessarily double first‑year profits or expenses. Scale efficiencies often defy linearity. Goal programming and other techniques account for non‑linear factors.
Linear programming delivers accurate results only when the model mirrors reality. Each model rests on assumptions that may be invalid. Assuming, for example, that tripling production will triple sales may overshoot market capacity, producing nonsensical outputs such as a recommendation to build 23.75 battleships for the Navy. Practitioners must adjust models to reconcile mathematical outcomes with practical feasibility.
Certain scenarios contain so many variables that a linear framework cannot capture them all. A medical practice might use linear programming to optimize radiation doses for cancer patients, yet individual patient variations often fall outside any linear model. Moreover, linear programming lacks intuition or gut instinct; as Heath Hammett noted in a 2005 interview with “Signal” magazine, human oversight is essential before implementing the results.
