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?
Understanding Operations Research
The hardest part of operations research is usually not the calculation. It is turning a messy situation into a model without losing the important details. Start by identifying choices that can change, such as the number of units to make or the route a driver takes.
Then list every limit separately. A factory may have limits on worker hours, machine time, raw materials, storage space, or money. Units must be consistent.
If one number is in minutes and another is in hours, convert one before building the model. A model is only as useful as its assumptions, so state them clearly. Linear models assume that each extra unit uses the same amount of a resource and produces the same added cost or benefit.
For a problem with two choices, a graph can show what the equations mean. Each boundary line represents using a resource exactly to its limit. The permitted area is where all the limits overlap.
Testing the corner points works because a straight objective line moves across this area until its last contact occurs at a corner. This rule depends on the problem being linear. If costs change at different production levels, or if making one product changes the profit of another, a straight-line model may not fit.
Students should check whether whole-number answers are needed. A plan to hire two point four workers may be mathematically valid but impossible in practice. Such cases need integer programming or a careful comparison of nearby whole-number plans.
Unused capacity carries information. If a resource has a large amount left over at the best plan, increasing that resource may not improve the result. A resource used completely may be a bottleneck.
Managers often want to know the value of one more hour of labor or one more kilogram of material. This is the idea behind sensitivity analysis. It tests how much an answer changes when prices, available resources, or requirements change.
Real data is rarely exact. Demand forecasts can be wrong, delivery times can vary, and costs can rise.
Good decision making does not treat one calculated answer as permanent. It compares several realistic cases and looks for a plan that remains workable.
Network models represent places as nodes and connections as edges. A shortest path method adds the cost along each possible route, but the cheapest route is not always the route with the fewest stops. The edge value must match the goal.
An ambulance route may use travel time. A delivery company may use fuel cost. A computer network may use delay.
Decision analysis handles a different kind of uncertainty. Outcomes may depend on weather, customer demand, or equipment failure. Expected value combines each possible payoff with how likely it is.
It gives a long-run average, not a promise for one event. A choice with the highest expected value can still have a serious risk, so students should pay attention to worst-case outcomes, probabilities, and the consequences of being wrong.