Business Optimization
Key points
The Shape of a Linear Program
Every problem of this kind has exactly two parts: an objective function (the thing you want to make as large or as small as possible) and constraints (the rules you cannot break). If all of it is linear — variables appear only to the first power — it is a linear program, and it can be solved. For example: maximize profit = ¥30,000 × (units of A) + ¥50,000 × (units of B), given no more than 200 kg of material and no more than 100 machine hours. Excel's Solver or a few lines of Python will find the best answer in seconds.
Writing Down the Objective and the Constraints
Let x₁ and x₂ be the decision variables — how many of each thing you make. Write the objective on one line and each constraint below it. The feasible region is the set of points that satisfy every constraint, and the best answer always sits at one of its corners. That is the fundamental theorem of linear programming, and it is why you only ever have to check the corners.
The Transportation Problem
You have goods sitting in several warehouses and customers in several places who need them. What shipping plan costs the least? Here x_{ij} is how much you ship from source i to destination j, and c_{ij} is what one unit on that route costs. No warehouse can ship more than it holds, and every customer must get what they ordered. This one model sits at the heart of supply chain and logistics planning.
Sensitivity Analysis: How Much Slack Is in the Answer?
Getting the optimal answer is only half the job. The question that matters next is: what happens if a constraint moves a little? Suppose you could raise machine time from 100 hours to 110 — how much more profit would that buy you? That number is the shadow price, and it tells you exactly what loosening a constraint is worth. Compare it to what the extra capacity costs and you have your answer on whether to invest. Excel's Solver produces all of this automatically in its sensitivity report.
Scheduling and Combinatorial Optimization
Staff scheduling, project planning with the critical path method, and the traveling salesman problem can all be written as integer programs, where the variables must be whole numbers or plain yes/no choices. That small change makes the problem enormously harder: the work required explodes as the problem grows. So in practice people reach for heuristics — methods that find a very good answer without proving it is the best — or for dedicated solvers like Gurobi and OR-Tools. The real goal is not perfection; it is a good answer in the time you actually have.
Jobs that use this
Operations Research Analyst$160k
Supply Chain Consultant$140k
Production Manager$110k
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