Business Applications of Operations Research

1. 1Name the decision's objective in one sentence before touching any model, using the passage's list of objectives: maximize profit, minimize cost, improve efficiency, throughput, recovery, or customer service. The objective function in the model must be this sentence expressed in algebra.
2. 2Pick one primary KPI that matches the objective (profit, cost saved, market share, or an output-per-input efficiency ratio) and fix its unit and period. Examples from the passages: € saved over 2008–2011, additional annual profit in €, $ saved over 2007–2010.
3. 3Record the baseline before the decision goes live: current cost, profit, queue length, or output per unit of input. Without a baseline, 'savings' cannot be computed.
4. 4Define the model's constraints from the real limits (budget, capacity, time slots, availability). Performance claims are valid only if the solution is feasible in operations, so check that the model reflects those limits.
5. 5Solve and record the model's predicted KPI value (for example, the objective value or the expected queue length). This is your forecast of business performance.
6. 6Implement the solution, then measure the same KPI over the same defined period and compare it to the baseline and the forecast.
7. 7For service systems, translate the decision into performance numbers a customer feels. Compute average queue length and average waiting-plus-service time for each candidate service rate, then choose the rate that meets the target.
8. 8For multi-output operations where a single ratio is hard (freight ton-km plus passenger-km per manpower), use an efficiency method such as Data Envelopment Analysis so units compare fairly, and report relative efficiency as the KPI.

Queue performance at a sales counter shows how a decision maps to a measurable outcome. The counter serves an average of 120 customers per hour, so the service rate μ is 2 per minute. Customers arrive at λ = 1.5 per minute. Because μ exceeds λ, the formulas apply: the average number in queue is λ²/(μ(μ−λ)) = 2.25, and the average waiting-plus-service time is 1/(μ−λ) = 2 minutes. The service-time distribution gives P(t ≤ 2 minutes) = 0.98, P(t ≤ 1 minute) = 0.86, and P(t ≤ 0.5 minute) = 0.63. The decision lever is μ. Raising it for the expected λ shortens the queue and the wait, and you re-run the two formulas for each μ and λ pairing to compare KPI outcomes. The baseline is 2.25 in queue and 2 minutes, and the post-change measurement is compared against both the baseline and the model's forecast.

• If the objective is cost or profit, then report the KPI in currency over a stated period, as Netherlands Railways did (€40 million additional annual profit) and TNT Express did (€207 million saved, 2008–2011).
• If service rate μ is at or below arrival rate λ, then the queue formulas do not apply; raise μ above λ before comparing performance.
• If outputs have different units (ton-km versus passenger-km), then do not add them; use Data Envelopment Analysis to compare efficiency.
• If a result is reported without a baseline and a time window, then treat it as unverified and do not count it as business performance.
• If the model has several goals (cost, robustness, utilization), then name which KPI is primary and track the others as secondary, as the railway timetable did with profit alongside utilization and robustness.

• Measuring model output instead of realized outcome: the report quotes the optimizer's objective value, but no post-implementation KPI exists. The tell is the absence of a before/after comparison.
• No baseline: the savings figure has nothing to be subtracted from. The tell is a claim like 'improved efficiency' with no starting number.
• Unbounded or mismatched period: the KPI mixes a one-time saving with an annual profit. The tell is a figure with no start or end year.
• Ignoring feasibility: the solution assumes capacity that does not exist, so the predicted gain never appears. The tell is constraints that were never checked against real limits.
• Adding incomparable outputs into one ratio: freight ton-km and passenger-km are summed to compare railroads. The tell is an efficiency score that changes sharply with the unit chosen.

• ▸Decision: describe the decision in one sentence.
• ▸Objective: state what is maximized or minimized (profit, cost, efficiency, service).
• ▸Primary KPI and unit: name the one measure and its unit (€, $, minutes, output per input).
• ▸Baseline value: record the KPI before implementation.
• ▸Measurement period: give start and end dates for the baseline and for the post-change reading.
• ▸Model-forecast KPI: enter the value the model predicts.
• ▸Realized KPI: enter the measured value after implementation.
• ▸Gap (realized minus forecast): record the difference and its likely cause.
• ▸Secondary KPIs tracked: list other measures, such as utilization or robustness, with their values.

1. 1State the decision in one sentence that begins with a verb and names the quantity you control. In the advertising problem: 'determine the number of minutes the advertisement should be aired on TV1 and TV2.' If the sentence names no controllable quantity, you have a goal rather than a decision.
2. 2Name one decision variable per controllable quantity and fix its unit. For example, x1 = minutes of advertising on TV1 and x2 = minutes on TV2. Write the unit beside the symbol so cost and viewership coefficients stay consistent.
3. 3Write the objective as one function of those variables and mark it maximize or minimize. The source states the aim as 'the number of viewers is maximized'. Viewership is 2 million per minute on TV1 and 3 million on TV2, so the objective is Max 2x1 + 3x2 (million viewers).
4. 4Confirm the linearity assumption before using it. The source relies on the company's finding of a linear relationship between airtime and viewers: 3 minutes gives 6 million on TV1 and 9 million on TV2. If doubling the input does not double the output, the problem is not linear programming as written.
5. 5List every limit as a constraint in the same units as the objective's inputs. Budget: 100,000x1 + 200,000x2 <= 1,000,000 GMD. Slot caps: x1 <= 3 and x2 <= 4 minutes. Add non-negativity (x1, x2 >= 0) explicitly.
6. 6Check each constraint against the problem statement by tracing it to a stated fact: budget, channel slot limit, or physical impossibility. Delete any constraint you cannot trace. Add any stated limit you have not yet written down.
7. 7Test the structure with a few hand-picked combinations before solving. The source lists several airtime mixes. Compute the cost and viewership of each to confirm that your constraints accept feasible mixes and reject infeasible ones.
8. 8Enter the formulation in software such as LINGO only after the variables, objective, and constraints are written on paper. The software solves the model but cannot tell you whether it matches the business problem.

A company launches a product and plans TV advertising on two channels, TV1 and TV2, with a budget of 1,000,000 GMD. An advertisement costs 100,000 GMD per minute on TV1 and 200,000 GMD per minute on TV2. Slots last at most 3 minutes on TV1 and 4 minutes on TV2. Viewership is 2 million per minute on TV1 and 3 million per minute on TV2, assumed linear. Decision variables: x1 and x2, the minutes aired on TV1 and TV2. Objective: maximize 2x1 + 3x2. Constraints: 100,000x1 + 200,000x2 <= 1,000,000; x1 <= 3; x2 <= 4; x1, x2 >= 0. Spot check: x1 = 3 and x2 = 3 costs 300,000 + 600,000 = 900,000 GMD, which is within budget, and yields 6 + 9 = 15 million viewers. Spot check: x1 = 3 and x2 = 4 costs 1,100,000 GMD and breaks the budget. Writing the model this way turns a vague question about how to spend the budget into a problem a solver can answer.

• If the decision sentence contains no quantity you can set, then keep refining it before writing any equations.
• If a stated limit (budget, capacity, slot length) is missing from your constraint list, then add it, because the optimizer will exploit any limit you leave out.
• If the relationship between a decision and its payoff is not proportional, then do not force a linear model. Choose a different technique, such as integer or non-linear programming, or revise the assumption.
• If you cannot trace a constraint to a stated fact, then drop it, since an invented constraint shrinks the feasible region and hides better solutions.
• If two stakeholders want different things optimized, then pick one objective for the first model and treat the other as a constraint or a separate multi-objective model.

• Writing a goal instead of a decision. The tell: the variables list is empty or holds outcomes such as 'viewers' rather than controllables such as 'minutes'.
• Mixing units across the model. The tell: budget in GMD sits beside coefficients in thousands or millions without conversion, and the solution looks off by orders of magnitude.
• Omitting a stated limit. The tell: the solution recommends x2 = 5 minutes when the channel caps slots at 4, or spends more than the budget.
• Assuming linearity without checking it. The tell: the plan treats the 10th minute of airtime as worth exactly the same as the first, with no evidence from the business.
• Opening the software before the problem is structured. The tell: the model solves cleanly but nobody can say which business fact each constraint represents.

• ▸Decision sentence: one verb-led sentence naming the quantity you control.
• ▸Decision variables and units: one symbol per controllable quantity, with its unit (e.g., x1 = minutes on TV1).
• ▸Objective: the single function to maximize or minimize, written with coefficients and units.
• ▸Objective direction: Max or Min, and the business measure behind it.
• ▸Constraints list: each limit as an inequality or equation, in consistent units.
• ▸Constraint source: the stated fact each constraint traces to (budget, cap, capacity).
• ▸Non-negativity and integrality: whether variables can be fractional or must be whole numbers.
• ▸Linearity check: evidence that output scales proportionally with each decision variable.
• ▸Spot-check combinations: two or three hand-computed mixes, noting feasible or infeasible.

1. 1List every resource the decision draws on and name the unit for each. In the source example the resources are advertising budget (thousand GMD), slot length on TV1 (minutes) and slot length on TV2 (minutes). Add labor, materials or capacity only if the problem actually has them.
2. 2Write each limit as a documented number with its source. The budget is 1,000 thousand GMD. TV1 and TV2 have informed the company that slots last at most 3 and 4 minutes. If you cannot point to who set a limit, treat it as an assumption and flag it.
3. 3Define decision variables in the same unit the limits use. Here x is minutes on TV1 and y is minutes on TV2, so cost per minute (100 and 200 thousand GMD) converts them into budget use.
4. 4Convert each limit into an inequality of the form (technological coefficients × variables) ≤ RHS. Budget: 100x + 200y ≤ 1000. Slot caps: x ≤ 3 and y ≤ 4. The RHS is the quantity of resource available, and the coefficients are the resource consumed per unit of each variable.
5. 5Add the implicit limits: non-negativity (x ≥ 0, y ≥ 0), plus integer or binary restrictions if the real world demands them (for example, slots only in whole minutes). Each restriction changes the model type, so state it deliberately.
6. 6Test your constraint set against rejected candidate plans. Combination 2 broke the budget and Combination 3 broke the slot cap. If a plan you know is infeasible still passes, a constraint is missing.
7. 7Solve, then read slack and dual prices for every constraint. Slack is the resource left over, and a dual price is the objective gain per extra unit of the resource. Zero slack means the constraint binds.
8. 8Rerun with changed RHS values to confirm the dual price and see how far the feasible space moves when a limit is relaxed or tightened.

The source problem maximizes viewership 2x + 3y, where x and y are minutes of advertising on TV1 and TV2. The resources are a 1,000 thousand GMD budget, with TV1 at 100 and TV2 at 200 thousand GMD per minute, and maximum slot lengths of 3 minutes on TV1 and 4 minutes on TV2. The constraints are 100x + 200y ≤ 1000, x ≤ 3, y ≤ 4, and x, y ≥ 0. The LINGO solution gives an objective of 16.5 with y = 3.5. The slack on constraint 3 (y ≤ 4) is 0.5, meaning 0.5 minute of TV2 slot capacity is unused, so that cap does not bind. The dual price on the budget constraint is 0.015. Raising the budget from 1000 to 1001 lifts viewership to 16.515, and raising it to 1003 lifts it to 16.545. Cutting it to 999 drops it to 16.485. The budget is therefore the binding resource, and each extra thousand GMD is worth 0.015 viewership units.

• If a constraint shows zero slack and a positive dual price, then that resource is binding. Spend analysis effort on it and value any extra unit at the dual price.
• If a constraint shows positive slack, as with the TV2 slot cap (slack 0.5), then relaxing it adds nothing. Look elsewhere for improvement.
• If a candidate plan breaks a known limit, then add or tighten a constraint until the model rejects it, as the budget and slot-length constraints do for Combinations 2 and 3.
• If a resource must come in whole units (whole minutes, whole machines), then add integer or binary restrictions and accept that the model is now an integer program rather than a plain LP.
• If a resource quantity is not known with certainty, then use a stochastic model instead of a single RHS value. Also run sensitivity checks on the RHS.

• Omitting an implicit limit such as non-negativity or a slot cap. The tell is a solution that is mathematically optimal but impossible in practice, such as a negative number of minutes or a slot longer than the channel allows.
• Mismatched units between coefficients and RHS (GMD vs. thousand GMD). The tell is a budget constraint that is either never binding or wildly infeasible.
• Treating an assumed limit as documented. The tell is that nobody can name who set the number, and the optimal plan shifts sharply when you change it.
• Ignoring dual prices and slack. The tell is spending time debating a non-binding constraint, such as the 0.5-minute TV2 slack, while the binding budget goes unexamined.
• Ignoring integrality or software limits. The tell is fractional answers (y = 3.5) where whole units are required, or a model that exceeds solver limits (the demo LINGO allows 500 variables and 250 constraints).

• ▸Resource name: the budget, time, capacity, labor or material being limited.
• ▸Unit: the unit of measure, such as thousand GMD or minutes.
• ▸Limit (RHS): the available quantity as a number.
• ▸Source of limit: who or what document sets it, or 'assumption'.
• ▸Consumption per unit of each decision variable: the technological coefficients, such as 100 per minute on TV1.
• ▸Constraint inequality: the written form, such as 100x + 200y ≤ 1000.
• ▸Integer or binary requirement: yes or no, and for which variables.
• ▸Slack at optimum: the unused resource reported by the solver.
• ▸Dual price: the objective gain per extra unit of the resource, read from the solver.

1. 1Write the objective as one sentence with a verb, a quantity, and a unit: 'maximize profit', 'minimize capital cost plus travelling cost', 'maximize people within 30 miles'. If you cannot name the quantity, you do not yet have an objective.
2. 2Say whether the quantity is maximized or minimized, then convert it into a formula over decision variables. In the product-mix case, profit per unit f_i times units x_i summed over four products gives 50x1 + 100x2 + 30x3 + 27x4. Every term must carry the objective's own unit (profit, cost, distance).
3. 3Check that the formula matches the sentence. The product-mix text asks 'how many units… to maximize the demand' but then defines and models profit. The model is right and the sentence is wrong. Fix the sentence before anyone reads results.
4. 4List the constraints separately and keep them out of the objective. Machine hours (4x1+2x2+x3+2x4 ≤ 100 and so on) and the demand cap (x2 ≤ 3) limit the solution. They are not goals. If a limit is really something you want more or less of, it belongs in the objective.
5. 5If more than one goal competes, decide the trade-off explicitly. Either fold them into one measure with stated rates, as in the supermarket cost model (capital cost plus 80 GMD per person per mile), or weight them. For weighting, run pairwise comparisons on a 1-9 scale, normalise each column, and average each row to get weights that sum to 1.
6. 6Solve with the objective and constraints in separate lines of the model. Then read the sensitivity output. A shadow price (objective changes by 0.015 per unit of budget) tells you what relaxing a limit is worth. A reduced cost (13 for x) tells you how much a variable's coefficient must improve before it enters the solution.
7. 7Re-test by changing the objective, not the constraints. The supermarket problem keeps the same three constraint families (at most 2 sites, each town assigned once, assign only to open sites) while the objective switches from 'maximize population served' to 'minimize total cost'. If the answer shifts, the objective is what drives the decision, so confirm stakeholders actually want that one.
8. 8Reconcile the result against the stated need. The supplier model says the company needs 800 pieces but writes the constraint as ≤ 800. Confirm the solution orders what the objective statement requires.

A company must order 800 pieces from four suppliers (S1–S4) and cares about price, quality, delivery consistency and process capability. Choosing on price alone would hide poor performance on the other three, so the objective is a weighted score, not a single measure. Stakeholders rate the criteria pairwise on a 1-9 scale (quality is rated seven times price; a41 = 5). Each column is normalised (a41 becomes 5/17) and the rows are averaged, giving weights of 0.0538 (price), 0.5749 (quality), 0.2244 (delivery) and 0.1470 (process capability), which sum to 1. The same method scores suppliers within each criterion. S1 scores 0.57, 0.17, 0.14 and 0.40, so its final score is (0.0538×0.57)+(0.5749×0.17)+(0.2244×0.14)+(0.1470×0.4) = 0.2186. The final scores are S1 0.2186, S2 0.4046, S3 0.1921 and S4 0.1885. The integer program then maximizes the sum of s'_j × l_j × x_j, where x_j is lots ordered. It is bounded by the 800-piece requirement and each supplier's maximum (400, 400, 600, 300) and lot sizes (40, 20, 100, 50). The stakeholder-set weights, not the solver, decide that quality dominates, so the weight session is where the real decision is made.

• If you can only state the goal as a single measure (profit, cost, distance), then write it directly as a linear sum of variable × unit value. Do not bring in weights.
• If the goal is genuinely several competing criteria, then weight them with pairwise comparison and check that the weights sum to 1 before building the model. Selecting on one criterion alone risks poor performance on the others.
• If two stakeholders could define 'best' differently (most people served versus least total cost), then build both objectives on the same constraints and compare the answers before committing.
• If the optimum leaves a variable at zero that you expected to be used, then read its reduced cost. It is the improvement in that variable's coefficient needed to bring it in (13 in the example).
• If a resource limit is binding, then read its shadow price before spending to relax it. A 0.015 gain per extra budget unit tells you whether the extra budget is worth it.

• The objective is described in words that differ from the formula. The tell is a problem statement saying 'maximize the demand' while the model maximizes profit. Re-read the sentence against the objective line.
• A limit is treated as a goal, or a goal as a limit. The tell is a demand cap (x2 ≤ 3) or hours cap sitting in the objective, or an unconstrained 'target' that the model ignores. Sort every item into 'maximize/minimize' or 'must satisfy'.
• One criterion is optimized alone and the others are ignored. The tell is the winner (cheapest supplier) ranking poorly on quality, delivery or process capability. Use weighted multi-criteria scoring.
• Weights are guessed, not elicited. The tell is that nobody can say where the importance ratings came from. Hold a structured session with stakeholders, record the pairwise comparisons, and normalise them.
• Unit mismatch in a combined cost objective. The tell is capital cost (GMD) added to travel cost without a per-person-per-mile rate. State the conversion (80 GMD per person per mile) in the formula.
• The statement of need and the constraint disagree. The tell is a required quantity of 800 modelled as ≤ 800. Check the solution total against the stated requirement.

• ▸Decision to be made: one sentence naming what is being chosen (products, sites, suppliers, quantities).
• ▸Objective statement: verb + quantity + unit (e.g., 'maximize profit in currency units').
• ▸Direction: maximize or minimize, and why that direction fits the business aim (profit/revenue up; cost/risk/distance down).
• ▸Objective formula: coefficient × decision variable for each term, with units on every coefficient.
• ▸Competing criteria: list any other goals; for each, state how it enters (merged by a rate, or weighted).
• ▸Weights and source: pairwise ratings on a 1-9 scale, who gave them, and confirmation that normalised weights sum to 1.
• ▸Constraints (not goals): resource limits, demand caps, required totals, lot sizes, supplier maxima.
• ▸Wording check: does the problem statement describe the same quantity the formula optimizes? Y/N and correction.
• ▸Sensitivity read-out: shadow prices of binding limits and reduced costs of unused variables, with the decision each implies.

1. 1State the decision in one sentence and name the objective: what is maximized or minimized (profit, cost, viewers, throughput). The source frames OR as decision making under scarce resources, so name the scarce resource too.
2. 2List the decision variables as the quantities you control, with units. In the advertising problem these are minutes of airtime on TV1 and on TV2.
3. 3Write every limit as a constraint using only supplied numbers: budget, per-unit cost, capacity caps, minimum requirements. Check that units match (GMD per minute against a GMD budget).
4. 4Test whether linearity holds. The advertising example is valid LP because the company found a linear relationship between airtime and viewers (2 and 3 million per minute). If a quantity must be whole (a plant on or off), use integer programming, as MISO did with mixed integer programming. If the relationship is nonlinear, you need a different tool, such as the nonlinear programming used by the Dutch Delta Program.
5. 5Enter the model in solver software (the source uses LINGO, with a limited free version at lindo.com). Keep the objective, the constraints and the data separate so you can change inputs without rewriting the logic.
6. 6Solve and read the values of the decision variables, not just the objective value. Check each against every constraint by hand and see which constraints are binding.
7. 7Run sensitivity and scenario analysis: change the budget, caps or per-unit yields and re-solve. The source says the model's value includes understanding the system better, not only getting one answer.
8. 8Translate the result back into the business decision, with the assumptions (linear yield, fixed prices, caps) stated alongside it.

Source data: a company has a 1,000,000 GMD budget to advertise on TV1 and TV2. Airtime costs 100,000 GMD per minute on TV1 and 200,000 GMD per minute on TV2. Slots are capped at 3 minutes on TV1 and 4 minutes on TV2. Viewership is 2 million per minute on TV1 and 3 million per minute on TV2, and the goal is to maximize viewers. Model: maximize 2x1 + 3x2 subject to 100,000x1 + 200,000x2 ≤ 1,000,000 (equivalently x1 + 2x2 ≤ 10), x1 ≤ 3, x2 ≤ 4, x1 and x2 ≥ 0. Working it out from these data: TV1 yields 2 million viewers per 100,000 GMD and TV2 yields 1.5 million, so TV1 is filled to its cap first (x1 = 3, spending 300,000). The remaining 700,000 buys x2 = 3.5, which is within the 4-minute cap. The result is 6 + 10.5 = 16.5 million viewers, with the budget and the TV1 cap both binding. If slots had to be whole minutes (an assumption not in the source), the problem becomes integer programming. The best choice is then x1 = 2, x2 = 4 for 16 million, so the integer requirement costs 0.5 million viewers. The book's own solution is cut off in the supplied text, so these figures are my arithmetic on its data.

• If the objective and every constraint are linear in the decision variables, use linear programming. If some variables must be whole numbers or on/off, use integer or mixed integer programming.
• If the optimum uses fractional values (3.5 minutes) and the real decision is discrete, re-solve as an integer program rather than rounding by eye, and compare the two objective values.
• If a constraint is binding (budget, cap), run a scenario that relaxes it. The change in the objective tells you what extra resource is worth.
• If the relationship between inputs and outcome is not linear, or the objectives conflict, do not force an LP. Choose nonlinear or multi-objective programming, both named in the source.
• If the solver's answer looks counterintuitive, check the data and units before trusting or rejecting it.

• Assuming linearity without evidence: the model is optimal on paper but fails in practice. The tell is that nobody can state the observed relationship (like the 2 and 3 million per minute estimates) behind the coefficients.
• Omitting a real constraint, such as the slot caps: the solver puts everything on the cheapest option. The tell is an all-in answer that exceeds what the channel allows.
• Unit mismatch, such as GMD per minute against a budget in millions: the model is infeasible or absurdly loose. The tell is a result that spends a tiny or impossible fraction of the budget.
• Rounding a fractional solution instead of solving as an integer program: the result is feasible but suboptimal (16 vs 16.5 vs a rounded-down guess). The tell is that no one re-solved after rounding.
• Reporting one optimal answer without sensitivity or scenario runs: the plan breaks when the budget or yields shift. The tell is that no one can say which constraints are binding.

• ▸Decision to be made: one sentence naming the choice and the scarce resource.
• ▸Objective: what is maximized or minimized, in units.
• ▸Decision variables: each controllable quantity, with units and bounds.
• ▸Constraints: each limit as an inequality using sourced numbers (budget, caps, minimums).
• ▸Data and evidence for coefficients: where each rate or cost comes from, and whether linearity is justified.
• ▸Model type: LP, integer, mixed integer or other, with the reason.
• ▸Solver output: variable values, objective value and binding constraints.
• ▸Scenarios run: which inputs were changed and how the optimum moved.
• ▸Business decision and caveats: the action to take and the assumptions behind it.

1. 1Write the objective in one sentence with its direction (maximize viewers, minimize cost) before any algebra. Optimality only means something against a stated objective function, so every later check refers back to it.
2. 2Name the decision variables as the quantities you control, with units. In the source case, x and y are minutes of advertising on TV1 and TV2. State whether each may be fractional (LP) or must be integer or binary (integer programming, as in the distribution-center variables y_kl and z_k).
3. 3Turn every real-world limit into a constraint. Budget becomes 100x + 200y ≤ 1000, slot length becomes x ≤ 3 and y ≤ 4, and non-negativity becomes x ≥ 0 and y ≥ 0. A limit you leave out is one the solver ignores, and the answer will not respect it.
4. 4Label the parameters: objective coefficients, technological coefficients, and right-hand sides. The solution is optimal only for these values, so record where each came from.
5. 5Before solving, test a few hand-built candidate plans against the constraints, as the source's four combinations do. Classify each as infeasible, feasible but wasteful (budget unspent), or feasible. This gives you a benchmark and exposes constraints you forgot.
6. 6Solve with a solver matched to the model class: simplex, dual simplex, or barrier for LP, and integer methods when variables are binary or integer. Read the optimal variable values and the objective value together.
7. 7Verify the output yourself. Substitute the values into every constraint, recompute the objective by hand, and confirm that it beats every feasible candidate from step 5.
8. 8Run sensitivity and scenario analysis on the budget, slot limits, and coefficients. Report the recommendation as optimal under stated assumptions, with the range over which it holds, not as an unconditional best.

The source's advertising problem is: maximize 2x + 3y viewers (millions) subject to 100x + 200y ≤ 1000 (thousand GMD), x ≤ 3, y ≤ 4, and x, y ≥ 0. Table 1.1 lists four hand-built plans. Plan 1 (1,1) gives 5 million viewers but spends only 300 of 1000. Plan 2 (3,4) gives 18 million but costs 1100, so it is infeasible. Plan 3 (4,3) gives 17 million at cost 1000, but 4 minutes exceeds TV1's 3-minute limit, so it is infeasible. Plan 4 (3,3) is feasible with 15 million viewers but spends only 900. The two highest totals, 18 and 17, are both infeasible and therefore not candidates. Solving the model (my arithmetic on the stated parameters, not a figure given in the passages) gives x = 3, y = 3.5. Cost is 300 + 700 = 1000, viewers are 6 + 10.5 = 16.5 million, and every constraint holds. This beats Plan 4 by 1.5 million and is also better than the feasible alternative (2,4), which gives 16 million at cost 1000. The optimum is fractional (3.5 minutes), which is valid because LP allows fractional decision variables.

• If a candidate plan violates any constraint, discard it however high its objective value is (Plans 2 and 3 score 18 and 17 but are infeasible).
• If a feasible plan leaves a binding resource unspent (Plan 4 spends 900 of 1000), treat it as probably suboptimal and test whether the unused resource can raise the objective.
• If the decision variables must be whole units or yes/no choices (select distribution center k, assign customer l), declare them integer or binary instead of using LP. Otherwise the 'optimal' answer may be fractional and unusable.
• If two moves tie on the criterion (the cable-layout step where G–F and B–E have equal distance), treat either as acceptable and note that alternative solutions exist. Do not claim the chosen one is uniquely best.
• If a parameter such as the budget, a coefficient, or a capacity is uncertain, run sensitivity or scenario analysis before presenting the answer. Optimal for the model is not optimal for reality when inputs shift.

• Treating the solver's output as the best real-world decision rather than the best answer to the model. The tell: nobody can say which assumptions or constraints were simplified or omitted.
• Missing constraints. The tell: the optimum looks excellent but breaks a business rule (a slot longer than the channel allows, as in Plan 3), or someone says 'we can't actually do that'.
• Using the wrong model class. The tell: fractional answers for indivisible choices, or integer logic forced onto a variable that can legitimately be fractional.
• Reporting an objective value without checking feasibility. The tell: a headline number such as 18 million viewers that costs 1100 against a 1000 budget.
• Quoting a single optimum with no sensitivity analysis. The tell: a small change in the budget or a coefficient flips the recommendation, and no one examined that range beforehand.

• ▸Objective (direction + metric): State in one sentence what is being maximized or minimized.
• ▸Decision variables and type: List each variable with its unit and whether it is continuous, integer, or binary.
• ▸Constraints: Write each limit as an inequality with its right-hand side and source.
• ▸Parameters and sources: List objective coefficients, technological coefficients, and RHS values, with where each came from.
• ▸Benchmark plans: List 3–4 hand-built plans, each with its objective value and a feasible/infeasible verdict.
• ▸Solver and method: Name the model class (LP, integer) and the algorithm or software used.
• ▸Solver result: Record the variable values and the objective value.
• ▸Manual check: Substitute the values into every constraint and recompute the objective. Mark pass or fail.
• ▸Sensitivity range: Record how far the budget or key coefficients can move before the plan changes, and note any ties or alternative optima.