A workshop can allocate a limited resource between two activities. Let x and y be nonnegative quantities, with x+y≤5, and suppose the synthetic benefit is 3x+2y. Before solving, ask whether fractional quantities are meaningful and whether benefit is really linear over the allowed range.
The model consists of the feasible set and objective together. An attractive objective value at an infeasible point is not an achievable decision.
Predict
Predict the optimum without enumerating every point. Can you bound 3x+2y using the resource inequality? Then identify a feasible point that attains your bound.
Experiment
Because y≥0, 3x+2y≤3x+3y≤15. The point (5,0) attains 15.
Proposal
Resource use
Benefit
Feasible?
(5,0)
5
15
Yes
(2,3)
5
12
Yes
(4,2)
6
16
No
Now add a minimum requirement y≥2. The earlier optimum is no longer feasible. Rewrite the objective as 3(x+y)−y≤15−2=13, attained at (3,2). A changed constraint requires a new certificate, not merely a new label on the old answer.
Understand
An optimization model should name decision variables, units, constraints, objective and uncertainty interpretation. The optimization course provides general duality and algorithmic machinery. In modeling, the additional obligation is to justify why those variables and constraints represent feasible actions and why the objective represents the stakeholder’s purpose.
Expected-loss decisions use probabilities over scenarios. Worst-case decisions minimize the largest loss over a specified uncertainty set. Regret compares a chosen action with the best eligible action in each scenario. These are different criteria, and none is universally appropriate. State the loss table and information available before selecting one.
For example, action A may cost 2 in a routine scenario and 20 in a disruption, while B costs 8 in either. Expected cost favors A when disruption probability is below 1/3, while worst-case cost favors B. The disagreement comes from the criterion, not an arithmetic error. If the probabilities are poorly supported, sensitivity to that threshold belongs in the report.
Continuous relaxations can give useful bounds for integer decisions. However, rounding can violate capacities, equalities or logical conditions. Check the rounded proposal explicitly. For maximization, the relaxed optimum is an upper bound on the integer optimum, while any feasible integer proposal supplies a lower bound. Reversing these directions would invalidate an optimality claim.
Apply
Formulate a two-activity allocation with two resources and a stated minimum service requirement. Produce one feasible decision and one general bound. Then compare expected and worst-case choices under two explicit scenarios. Explain which conclusion remains stable when a parameter varies and which depends on an uncertain threshold.
Continue
Simplification can make a difficult model analyzable, but it needs a regime and error argument. Perturbation and similarity methods next show how a reduced formula can be useful without pretending to be exact everywhere.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Maximize 2x+y with x,y≥0 and x+y≤4. What is the optimal value?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A chosen action costs 9 in a scenario where the cheapest eligible action costs 6. What is its regret?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Formulate a resource allocation with one objective and two resource constraints.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Certify the optimum of max 2x+y under x+y≤4 and nonnegativity without enumeration.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Compare minimizing expected loss with minimizing worst-case loss over stated scenarios.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Why can rounding a continuous allocation invalidate a decision?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
