Operations research uses math to help people choose the best action when resources are limited. This cheat sheet covers how to define decisions, build constraints, compare tradeoffs, and optimize results. Students need these tools for real problems in scheduling, routing, production, budgeting, and planning.
Key Facts
- An optimization model has decision variables, an objective function, and constraints.
- A linear programming objective has the form maximize or minimize Z = c1x1 + c2x2 + ... + cnxn.
- A linear constraint has the form a1x1 + a2x2 + ... + anxn <= b, >= b, or = b.
- The feasible region is the set of all solutions that satisfy every constraint, including nonnegative restrictions such as x >= 0 and y >= 0.
- For a two-variable linear programming problem, the optimal solution occurs at a corner point of the feasible region if an optimum exists.
- Slack in a <= constraint is slack = right side - left side, and it measures unused resource capacity.
- Expected value is EV = sum of probability × payoff, and it helps compare choices under uncertainty.
- In network problems, the shortest path minimizes total distance, time, or cost along connected edges.
Vocabulary
- Operations Research
- Operations research is the use of mathematical models to make better decisions in complex systems.
- Decision Variable
- A decision variable is a quantity the decision maker can choose, such as the number of units to produce.
- Objective Function
- An objective function is the formula being maximized or minimized, such as profit, cost, time, or distance.
- Constraint
- A constraint is a limit or requirement that every possible solution must satisfy.
- Feasible Region
- The feasible region is the set of all possible solutions that satisfy all constraints at the same time.
- Optimal Solution
- An optimal solution is the feasible choice that gives the best value of the objective function.
Common Mistakes to Avoid
- Mixing up maximize and minimize is wrong because the same feasible region can give different best choices depending on the goal.
- Forgetting nonnegative constraints is wrong because negative production, negative hours, or negative shipments usually have no real-world meaning.
- Graphing an inequality boundary correctly but shading the wrong side is wrong because it changes the feasible region and may produce impossible solutions.
- Choosing a point inside the feasible region without checking corner points is wrong because linear programming optima occur at vertices when an optimum exists.
- Ignoring units is wrong because adding hours to dollars or comparing unlike quantities makes the model meaningless.
Practice Questions
- 1 A company makes x chairs and y tables. Profit is P = 30x + 50y. If x = 12 and y = 8, what is the total profit?
- 2 For the constraint 2x + 3y <= 60, find the slack when x = 15 and y = 8.
- 3 A delivery network has route costs A to B = 4, B to D = 7, A to C = 6, and C to D = 3. Which path from A to D is shorter, A-B-D or A-C-D?
- 4 Why is it important to list decision variables before writing the objective function and constraints?