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INFORMS CAP study materials
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INFORMS CAP practice questions

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Question 1Business Problem Framing

A team is starting a CAP analytics project and needs to state the business problem before any modeling begins. Which phrasing best qualifies as a well-framed business problem statement?

  1. Reduce customer churn in the mid-market segment by 15% within four quarters by targeting at-risk accounts with retention offers Correct answer.
  2. Improve the churn model's AUC from 0.78 to 0.85
  3. Build a gradient boosting classifier with 200 trees on the churn dataset
  4. Explore customer data to find interesting churn patterns

Explanation

A well-framed business problem names the outcome, the affected population, a measurable target, and a time horizon, which lets success be judged by business results. Modeling choices and metric improvements belong to the analytical translation, not the business framing.

Question 2Business Problem Framing

In analytics project framing, what distinguishes a business problem from an analytical problem?

  1. The business problem is stated in terms of organizational outcomes and decisions, while the analytical problem specifies the data and methods needed to support them Correct answer.
  2. The business problem is written by executives and the analytical problem is written by software engineers
  3. The business problem concerns revenue only, while the analytical problem concerns cost only
  4. The business problem is always qualitative, whereas the analytical problem is always quantitative

Explanation

Business problems are framed around decisions, outcomes, and value for the organization; analytical problems translate them into measurable quantities and modeling tasks. The two are linked by a translation step that keeps analysis anchored to a decision.

Question 3Business Problem Framing

A stakeholder asks for 'a dashboard that shows everything about operations.' Before accepting the request, what is the most important framing step?

  1. Inventory every available data table so nothing is omitted
  2. Ask which decisions the dashboard should support and what actions would follow from the information Correct answer.
  3. Confirm the preferred visualization library and hosting environment
  4. Estimate the storage cost of retaining all operational logs

Explanation

Framing starts from decisions and actions, which determines what information is genuinely needed and how success will be judged. Tooling and data inventory are downstream choices that should follow from that purpose.

Question 4Business Problem Framing

Which element is essential when defining the scope of an analytics engagement?

  1. A guarantee that the first model will meet the target
  2. A fixed schedule with no allowance for data quality issues
  3. A complete list of every algorithm the team might try
  4. A clear boundary of what is included and excluded, agreed with the sponsor Correct answer.

Explanation

Scope is defined by explicit inclusions and exclusions agreed with the sponsor, which prevents expectation gaps later. Algorithm lists, performance guarantees, and rigid schedules are not scope statements and often create unrealistic commitments.

Question 5Business Problem Framing

A retail sponsor says the goal is 'to be more data-driven.' Which rewrite converts this into an actionable business problem?

  1. Appoint a chief data officer and stand up a governance council
  2. Train all staff on analytics tools within the year
  3. Adopt a modern data platform across the organization
  4. Increase repeat purchase rate among first-time buyers from 22% to 28% over two quarters Correct answer.

Explanation

An actionable problem names a measurable outcome, a target population, a quantified goal, and a timeframe. The other options describe initiatives or organizational changes whose success is not tied to a business result.

Question 6Business Problem Framing

What role does a hypothesis play in framing an analytics business problem?

  1. It replaces the need for a success metric
  2. It states a testable belief about how a driver affects an outcome, guiding what evidence to collect Correct answer.
  3. It guarantees the analysis will produce a positive result
  4. It fixes the modeling technique before any data review

Explanation

A hypothesis expresses a testable relationship between drivers and outcomes, which focuses data collection and analysis and makes findings falsifiable. It does not guarantee results, substitute for metrics, or lock in a method.

Question 7Business Problem Framing

An analytics team is asked to predict which loan applicants will default. Which framing question most directly affects how the business will use the model?

  1. How many features can the database store per applicant?
  2. Which programming language will the team use to build the model?
  3. What decision will the prediction change, and what are the costs of a false approval versus a false rejection? Correct answer.
  4. Should the model be retrained weekly or monthly?

Explanation

The decision context and the relative costs of errors determine the threshold, the metric, and the acceptable trade-offs in the model. Language, storage, and retraining cadence are implementation details that follow from that framing.

Question 8Business Problem Framing

Which statement best describes a stakeholder in the context of framing an analytics business problem?

  1. The vendor supplying the analytics software license
  2. Any employee who has read access to the analytics database
  3. A person or group who affects or is affected by the decision the analysis supports Correct answer.
  4. The individual who writes the production code for the model

Explanation

Stakeholders are those who influence or are impacted by the decision, which is why their interests shape problem framing and acceptance criteria. Database access, vendors, and developers may be involved but are not stakeholders by definition.

Question 9Business Problem Framing

A telecom sponsor wants to reduce subscriber churn but the data team can only observe customers who have already left. Which framing issue should be raised before proceeding?

  1. Whether the available data can represent the at-risk population and the timing needed for proactive intervention Correct answer.
  2. Whether churn should be measured as a rate or a count
  3. Whether the model should be logistic regression or a random forest
  4. Whether the dashboard should use monthly or quarterly aggregation

Explanation

Framing must confirm that data coverage and timing support the intended decision, since proactive retention requires signals available before departure. Metric definition, algorithm choice, and dashboard granularity are secondary and depend on that feasibility check.

Question 10Business Problem Framing

A sponsor asks for a model that predicts next quarter's revenue 'as accurately as possible.' Which reframing best supports a useful analytics engagement?

  1. Collect every available data source before defining the target
  2. Choose the most complex model available to maximize accuracy
  3. Specify the decision the forecast will drive, the acceptable error for that decision, and the planning horizon Correct answer.
  4. Commit to a forecast accuracy of 99% to satisfy the sponsor

Explanation

Accuracy only matters relative to a decision and an acceptable error tolerance, so framing should tie the forecast to its use and horizon. Promising arbitrary accuracy, hoarding data, or defaulting to complexity are not sound framing practices.

Question 11Business Problem Framing

During framing, a team maps the current process and finds that the 'churn prediction' request actually stems from slow response times in the retention call center. What is the most appropriate framing response?

  1. Reject the request and refer the sponsor to operations without further analysis
  2. Proceed with the predictive model because that is what the sponsor requested
  3. Surface the underlying operational issue and test whether analytics is the right lever before committing to a predictive model Correct answer.
  4. Expand the project to include call center staffing, forecasting, and marketing analytics

Explanation

Framing should test whether the stated request addresses the root cause and whether analytics is the appropriate intervention. Blindly building the model, dismissing the request, or ballooning scope all skip that diagnostic step.

Question 12Business Problem Framing

A team defines success for a fraud detection initiative. Which success measure is most appropriate at the business framing stage?

  1. Reduction in net fraud losses per quarter while holding customer friction complaints at or below current levels Correct answer.
  2. Area under the ROC curve of at least 0.90 on the holdout set
  3. Number of features engineered from transaction logs
  4. F1 score above 0.80 for the fraud class

Explanation

Business framing calls for outcome measures that reflect value and trade-offs, such as loss reduction balanced against customer friction. Model metrics and feature counts are analytical proxies that must be linked to such outcomes.

Question 13Business Problem Framing

Two sponsors disagree: one wants a churn model optimized to catch every at-risk customer, the other wants to avoid contacting satisfied customers. How should the analyst frame this conflict?

  1. Defer the decision until the model is built and results are available
  2. Make the trade-off explicit by quantifying the cost of missed churners versus unnecessary retention offers, and let the sponsor set the operating point Correct answer.
  3. Choose the first sponsor's objective because recall is generally more important
  4. Average the two objectives into a single unweighted score

Explanation

Conflicting objectives are a framing issue about error costs and risk tolerance that the business owner must resolve with quantified trade-offs. Picking a side, blending objectives arbitrarily, or postponing the choice leaves the model without a defensible target.

Question 14Business Problem Framing

Which practice best ensures that an analytics problem statement is testable?

  1. Describing the intended visual style of the final presentation
  2. Listing the tools and platforms the team will use
  3. Specifying the population, the outcome measure, the comparison baseline, and the time window Correct answer.
  4. Naming the executive who will receive the final report

Explanation

Testability requires a defined population, a measurable outcome, a baseline for comparison, and a time window so results can be verified. Tooling, report recipients, and presentation style do not make a claim verifiable.

Question 15Business Problem Framing

A team is framing an analytics project and must decide whether to build a descriptive, predictive, or prescriptive solution. What should drive that choice?

  1. The nature of the decision to be supported and the maturity of the underlying data and process Correct answer.
  2. The team's preference for the most advanced technique available
  3. The desire to publish novel methodology
  4. The vendor's recommended product roadmap

Explanation

The analytical approach should follow from what decision needs support and whether data and process maturity can sustain it. Technique preference, vendor roadmaps, and novelty are not valid drivers of problem framing.

Question 16Business Problem Framing

A sponsor insists the analysis must confirm that the new pricing policy increased margin. How should the analyst respond at the framing stage?

  1. Decline the project because confirmation is impossible
  2. Reframe the objective as estimating the policy's causal effect, acknowledging the result may show no increase Correct answer.
  3. Redefine the outcome as customer satisfaction to avoid the conflict
  4. Accept the objective as stated and select methods likely to produce a positive finding

Explanation

Framing must preserve objectivity, so the objective should be to estimate the effect honestly rather than to confirm a desired conclusion. Selecting methods for a predetermined result, refusing outright, or swapping the outcome all compromise the analysis.

Question 17Business Problem Framing

Which artifact most clearly captures the agreed framing of an analytics business problem for sign-off?

  1. A data dictionary listing every field in the warehouse
  2. A model card describing the final algorithm's performance
  3. A problem charter stating the decision, outcome metric, scope, stakeholders, and assumptions Correct answer.
  4. A project Gantt chart showing task dependencies

Explanation

A problem charter records the decision context, success metric, scope, stakeholders, and assumptions so sponsors can confirm shared understanding. Data dictionaries, model cards, and schedules document other aspects and do not capture the framing itself.

Question 18Business Problem Framing

An insurer wants to reduce claim processing time. The analytics team proposes starting with a process-mining view of where time is spent. Why is this consistent with good problem framing?

  1. It removes the need to involve claims staff in the project
  2. It guarantees that automation will be the chosen solution
  3. It avoids the need to define a success metric for processing time
  4. It establishes where variation and delay arise before committing to a predictive or prescriptive intervention Correct answer.

Explanation

Diagnostic analysis of where time is lost grounds later interventions in evidence and helps select the right analytical approach. It does not eliminate success metrics, predetermine automation, or remove the need for domain input.

Question 19Business Problem Framing

A team is scoping a customer segmentation project. Which statement correctly separates the business problem from the analytical problem?

  1. The business problem is to clean the CRM data; the analytical problem is to reduce missing values
  2. The business problem is to select a distance metric; the analytical problem is to normalize the features
  3. The business problem is to run k-means clustering; the analytical problem is to choose the number of clusters
  4. The business problem is to raise campaign response; the analytical problem is to derive stable segments that predict differential response Correct answer.

Explanation

The business problem concerns the outcome the organization wants, while the analytical problem specifies the modeling task and evidence needed to support it. The other options describe technical tasks mislabeled as business problems.

Question 20Business Problem Framing

Before approving an analytics project, a sponsor asks how value will be realized. Which framing element most directly answers this?

  1. A list of the data sources that will be integrated
  2. A value hypothesis linking the model's use to a measurable business outcome and the mechanism of change Correct answer.
  3. A description of the model's architecture and training procedure
  4. A summary of the team's prior project experience

Explanation

A value hypothesis explains how using the analysis changes decisions and produces measurable business benefit, which is what the sponsor needs. Data sources, architecture, and team experience describe inputs and capabilities rather than the path to value.

Question 21Business Problem Framing

A team frames a hospital readmission project. Which assumption, if left unstated, most threatens the project's validity?

  1. That readmissions recorded in the available data reflect the same patient population and time window the intervention will target Correct answer.
  2. That gradient boosting will outperform logistic regression
  3. That the project will finish within the fiscal year
  4. That the dashboard will refresh nightly

Explanation

Unstated assumptions about population and time window can invalidate conclusions if the data do not match the intervention's target context. Algorithm comparisons, refresh cadence, and schedule are secondary and do not undermine validity in the same way.

Question 22Business Problem Framing

A team is asked to 'reduce supply chain cost.' Which framing step best converts this broad goal into an analytics-ready problem?

  1. Model every cost driver simultaneously with a single complex model
  2. Decompose cost into drivers, identify which driver is addressable with available data, and set a target for that driver Correct answer.
  3. Estimate total supply chain cost and report it monthly
  4. Benchmark against competitors and adopt their analytics stack

Explanation

Decomposing the goal into addressable drivers and setting a targeted metric turns a broad aspiration into a tractable problem. Reporting totals, copying competitors, or modeling everything at once lacks the focus needed for a workable engagement.

Question 23Business Problem Framing

An analytics team proposes a real-time scoring system. A reviewer asks whether the latency requirement is justified by the decision it supports. Why is this a framing question rather than a purely technical one?

  1. Latency is always determined by the database vendor's specifications
  2. Latency requirements are set by the model's training time
  3. Real-time systems are inherently more accurate than batch systems
  4. Latency requirements follow from how quickly the business decision must be made and what value timely action creates Correct answer.

Explanation

Timeliness is a business requirement derived from decision speed and the value of acting sooner, so it belongs in problem framing. Vendor specs, accuracy claims, and training time do not establish why real-time operation is needed.

Question 24Business Problem Framing

A sponsor wants to measure the impact of a new recommendation engine but plans to launch it to all customers at once. What framing concern should the analyst raise?

  1. The engine should be built with a neural network to ensure measurable impact
  2. The launch should be delayed until all possible features are engineered
  3. Without a comparison group or a staged rollout, the effect of the engine cannot be separated from seasonal and market changes Correct answer.
  4. The impact should be measured by click-through rate alone

Explanation

Causal attribution requires a credible counterfactual, which a universal simultaneous launch removes, so the framing should propose a staged rollout or holdout. Algorithm choice, a single proxy metric, and feature completeness do not solve the attribution problem.

Question 25Business Problem Framing

A team frames a project to reduce equipment downtime. The sponsor wants a predictive maintenance model, but records show most downtime comes from scheduled maintenance overruns rather than unexpected failures. What is the strongest framing conclusion?

  1. The scope should expand to cover all plant operations to ensure relevance
  2. The predictive maintenance model should proceed because unexpected failures are the more interesting problem
  3. The problem definition should be revisited to target maintenance scheduling, with predictive failure modeling possibly deferred Correct answer.
  4. The project should be cancelled because predictive maintenance is not applicable

Explanation

Framing should follow the evidence about where the outcome is actually generated, which here points to scheduling rather than failure prediction. Proceeding on interest, cancelling, or expanding scope broadly ignores that diagnostic finding.

Question 26Methodology

A CAP candidate is structuring a decision problem in which the analyst must choose a production quantity before observing demand, and profit depends on both the quantity and the realized demand value. Which modeling framework is the standard fit for this single-stage choice under uncertainty?

  1. A Markov decision process with an infinite horizon
  2. A shortest-path network flow formulation
  3. A deterministic linear program with demand fixed at its mean
  4. A two-stage stochastic program with the quantity as a first-stage decision Correct answer.

Explanation

When a decision must be committed before an uncertain parameter is realized, the canonical model is a two-stage stochastic program: first-stage variables are chosen here-and-now, second-stage (recourse) variables respond after the random outcome. Fixing demand at its mean discards the distributional information that drives the hedging behavior.

Question 27Methodology

In a linear programming model, a constraint is binding at the optimal solution. What does the dual value (shadow price) associated with that constraint represent?

  1. The probability that the constraint remains feasible under parameter uncertainty
  2. The rate of change in the optimal objective value per unit relaxation of the constraint's right-hand side Correct answer.
  3. The reduced cost of the decision variable with the largest coefficient in that row
  4. The total number of units shipped along that constraint's arc

Explanation

The shadow price is the marginal value of relaxing the right-hand side by one unit, holding the basis fixed, and it is valid within the range over which that basis stays optimal. It is a sensitivity measure on the objective, not a count of flow or a probability.

Question 28Methodology

An analyst fits a multiple linear regression and reports a variance inflation factor of 12 for one predictor. Which interpretation is most defensible?

  1. That predictor is strongly collinear with the other predictors, so its coefficient is poorly identified Correct answer.
  2. That predictor's coefficient is statistically insignificant at the 5 percent level
  3. The residuals exhibit heteroskedasticity that must be corrected before inference
  4. The model's overall F test is invalid because the sample size is too small

Explanation

A variance inflation factor above roughly 10 signals that the predictor is largely explainable by the remaining predictors, inflating the standard error of its coefficient and making the individual estimate unstable. It says nothing directly about significance, residual variance structure, or the validity of the overall F test.

Question 29Methodology

A simulation study estimates the mean waiting time in a queue from 10 independent replications, each with a different random-number stream. Which procedure produces a valid confidence interval for the steady-state mean?

  1. Report the minimum and maximum replication averages as the interval endpoints
  2. Treat every individual customer waiting time across all replications as one pooled sample
  3. Compute the sample mean and standard error across the 10 replication averages and use a t critical value Correct answer.
  4. Use the single replication with the longest run length and apply a batch-means interval

Explanation

Replication averages from independent streams are independent and identically distributed, so the classical t-based interval on those averages is valid. Pooling individual observations ignores within-run correlation, and a single run requires an internal method such as batch means rather than cross-replication standard errors.

Question 30Methodology

In a decision tree for a capital project, a chance node is followed by a decision node. Which statement correctly describes the rollback procedure?

  1. Average at both nodes using equal branch weights
  2. Take the maximum at both nodes regardless of node type
  3. Average the payoffs at the chance node, then take the maximum at the decision node Correct answer.
  4. Take the maximum at the chance node, then average at the decision node

Explanation

Rollback folds the tree from the terminal branches back to the root: chance nodes are evaluated as expected values using branch probabilities, and decision nodes are evaluated by selecting the best available alternative. Reversing the operations at either node type yields the wrong value of the project.

Question 31Methodology

A manager must choose among three alternatives under four future scenarios and wishes to apply the maximin criterion. Which quantity is compared across alternatives?

  1. The payoff under the most likely scenario
  2. The worst payoff attainable under each alternative Correct answer.
  3. The difference between the best and worst payoff for each alternative
  4. The expected payoff using equal scenario probabilities

Explanation

Maximin is a conservative criterion that scores each alternative by its minimum payoff across scenarios and selects the alternative whose minimum is largest. It requires no probabilities and deliberately ignores the upside beyond the worst case.

Question 32Methodology

A transportation model has supplies of 40 and 60 units at two origins and demands of 30, 45, and 25 units at three destinations. Which statement about the model's standard form is correct?

  1. A dummy destination with demand 10 units must be added
  2. The model is infeasible because supply exceeds the smallest demand
  3. The model is balanced because total supply equals total demand of 100 units Correct answer.
  4. A dummy origin with capacity 25 units must be added

Explanation

The classical transportation problem is balanced when total supply equals total demand; here both equal 100 units, so no dummy node is needed. Balance is a property of the totals, not of any individual supply-demand pair.

Question 33Methodology

In a minimum-cost network flow model, which condition characterizes a negative-cost cycle in the residual network of a current feasible flow?

  1. It proves the current flow is optimal
  2. It shows that the problem has multiple optimal solutions
  3. It indicates that the supply and demand data are infeasible
  4. It identifies a way to reduce total cost by augmenting flow around the cycle Correct answer.

Explanation

Pushing flow around a residual cycle whose total cost is negative lowers the objective while preserving conservation and capacity feasibility, so the current flow cannot be optimal. Optimality holds exactly when the residual network contains no negative-cost cycle.

Question 34Methodology

A project network has an activity with earliest start 6, duration 4, and latest finish 14. What are the activity's total float and free float if its earliest finish successor starts at 12?

  1. Total float 8, free float 6
  2. Total float 4, free float 2 Correct answer.
  3. Total float 4, free float 4
  4. Total float 2, free float 4

Explanation

Total float equals latest finish minus earliest start minus duration, or 14 minus 6 minus 4, giving 4. Free float equals the earliest successor start minus earliest finish, or 12 minus 10, giving 2.

Question 35Methodology

In PERT probabilistic scheduling, the three-point estimate uses optimistic, most likely, and pessimistic durations. Which formula gives the standard PERT expected activity duration?

  1. (optimistic plus pessimistic) divided by 2
  2. (optimistic plus most likely plus pessimistic) divided by 3
  3. (4 times optimistic plus most likely plus pessimistic) divided by 6
  4. (optimistic plus 4 times most likely plus pessimistic) divided by 6 Correct answer.

Explanation

The PERT beta approximation weights the most likely value four times and the two extremes once each, dividing by six. This yields a mean that sits closer to the mode than the simple midpoint of the extremes.

Question 36Methodology

A queueing system has Poisson arrivals at rate 8 per hour and exponential service at rate 10 per hour with a single server. What is the probability that an arriving customer finds the server busy?

  1. 0.20
  2. 0.80 Correct answer.
  3. 0.64
  4. 0.10

Explanation

For the M/M/1 queue the utilization, and hence the steady-state probability that the server is busy, is the arrival rate divided by the service rate, or 8/10 = 0.80. Stability requires this ratio to be below one.

Question 37Methodology

An analyst is deciding between an M/M/1 model and an M/G/1 model for a service facility. Which feature of the real system most strongly motivates the M/G/1 choice?

  1. There is exactly one server
  2. Arrivals occur in a Poisson pattern
  3. Service times are not exponentially distributed Correct answer.
  4. The system operates in steady state

Explanation

The Pollaczek-Khinchine results for M/G/1 relax the exponential service assumption and depend on the service-time mean and variance, so non-exponential service is the decisive motivation. Poisson arrivals and a single server are shared by both models.

Question 38Methodology

A firm faces uncertain demand with mean 500 and standard deviation 80 and orders from a supplier with a fixed lead time. Under the newsvendor logic, what happens to the optimal order quantity as the underage cost rises relative to the overage cost?

  1. The order quantity becomes exactly the mean demand
  2. The critical fractile is unchanged because the demand distribution is fixed
  3. The critical fractile rises and the order quantity increases Correct answer.
  4. The critical fractile falls and the order quantity decreases

Explanation

The critical fractile equals the underage cost divided by the sum of underage and overage costs, so a higher underage cost raises the fractile. A higher fractile corresponds to a larger order because the optimal quantity is that quantile of the demand distribution.

Question 39Methodology

In an economic order quantity model, annual demand is 10,000 units, the fixed order cost is 50, and the annual holding cost per unit is 4. What is the economic order quantity?

  1. 2,000 units
  2. 500 units Correct answer.
  3. 1,000 units
  4. 250 units

Explanation

The economic order quantity equals the square root of twice the annual demand times the order cost divided by the holding cost, that is the square root of 2 times 10,000 times 50 divided by 4, which is the square root of 250,000, or 500 units.

Question 40Methodology

A discrete-event simulation uses a fixed time increment of one minute and updates the system state at each tick regardless of whether events occur. Which limitation is most characteristic of this approach?

  1. It may miss events that occur between ticks and incurs computation on idle intervals Correct answer.
  2. It cannot represent stochastic arrivals
  3. It cannot produce replication-based confidence intervals
  4. It requires the system to be Markovian

Explanation

Fixed-increment time advance examines the system at regular intervals, so events falling between ticks are handled inaccurately and many ticks produce no state change, wasting computation. Event-scheduling avoids both problems by advancing directly to the next event time.

Question 41Methodology

An analyst wants to reduce the variance of a simulation estimate of a system's mean performance without changing the number of replications. Which technique uses a correlated but analytically tractable companion system?

  1. Control variates using a known-expectation quantity Correct answer.
  2. Antithetic variates within each replication
  3. Common random numbers applied across competing system designs
  4. Stratified sampling of the input distribution

Explanation

A control variate exploits the correlation between the simulation output and an auxiliary quantity whose expectation is known, adjusting the estimator to remove part of the noise. Common random numbers target comparisons across designs, and antithetic variates use negatively correlated paired runs rather than a known-expectation companion.

Question 42Methodology

A Markov chain has a transition matrix in which every row is identical. Which statement about its long-run behavior is correct?

  1. The chain is periodic with period equal to the number of states
  2. The limiting distribution depends on the starting state
  3. The chain has no stationary distribution because states are transient
  4. The stationary distribution equals the common row vector, regardless of the initial state Correct answer.

Explanation

When each row of the transition matrix is the same distribution, the chain resets to that distribution after one step, so it is the stationary distribution and the limit is independent of the initial state. Such a chain is aperiodic because the diagonal entries are generally positive.

Question 43Methodology

In a Markov decision process, a policy is optimal for an infinite-horizon discounted problem. Which statement best characterizes it?

  1. It minimizes the number of transitions between states
  2. It maximizes the immediate reward at every state
  3. It maximizes the expected discounted reward from every state simultaneously Correct answer.
  4. It guarantees the highest reward along every possible sample path

Explanation

An optimal policy for the discounted infinite-horizon problem attains the maximal value function at every state, balancing immediate and future discounted rewards. Myopic rules and pathwise guarantees are strictly weaker or unattainable criteria.

Question 44Methodology

A nonlinear program maximizes a concave objective over a convex feasible region. Which property follows for a local maximum found by a gradient-based method?

  1. It may be improved by moving to any nearby feasible point
  2. It is optimal only if the objective is also differentiable everywhere
  3. It is a global maximum of the problem Correct answer.
  4. It is necessarily a saddle point rather than a maximum

Explanation

Maximizing a concave function over a convex set is a convex optimization problem in which every local maximum is global. This property is what makes gradient-based methods reliable for such formulations.

Question 45Methodology

A constraint in an integer program is described as a valid inequality. What does this mean?

  1. It is satisfied by every solution of the linear programming relaxation
  2. It is satisfied by every feasible integer solution but may cut off fractional points Correct answer.
  3. It guarantees that the relaxation has an integral optimum
  4. It tightens the objective function's coefficients

Explanation

A valid inequality does not exclude any feasible integer solution while potentially removing fractional points from the relaxation's feasible region. Cutting planes are valid inequalities used to strengthen the relaxation toward the integer hull.

Question 46Methodology

In a branch-and-bound procedure for a maximization integer program, a node's linear programming relaxation yields an objective value of 42 and the incumbent feasible integer solution has value 45. What is the appropriate action for that node?

  1. Branch on a fractional variable to create two child nodes
  2. Discard the incumbent and continue from this node
  3. Fathom the node because its relaxation cannot beat the incumbent Correct answer.
  4. Update the incumbent to 42

Explanation

For maximization, a node whose relaxation bound is no better than the incumbent cannot contain a better integer solution, so it can be pruned. This bound-based fathoming is what keeps the search tree manageable.

Question 47Methodology

A goal programming model contains several objectives with target levels and deviation variables. What does a preemptive priority structure imply about the solution process?

  1. Higher-priority goals are optimized first, and lower priorities are optimized only within the resulting optimal set Correct answer.
  2. All deviations are weighted into a single objective and minimized simultaneously
  3. Priorities are converted into monetary costs before solving
  4. Deviations are ignored when a goal is infeasible

Explanation

Preemptive goal programming solves a sequence of models in priority order, never allowing a lower-priority goal to degrade a higher-priority one. This lexicographic structure differs from weighted goal programming, where all deviations enter one objective.

Question 48Methodology

In data envelopment analysis, a decision-making unit is found to be efficient. What does this result mean within the model's assumptions?

  1. It produces the highest absolute output of any unit in the sample
  2. It is efficient regardless of which inputs and outputs are included
  3. It has the lowest cost per unit of output among all units
  4. No other unit or combination of units dominates it in the chosen input-output space Correct answer.

Explanation

Efficiency in data envelopment analysis is relative: a unit is efficient when no convex combination of observed units uses less input to produce at least as much output. Absolute output levels and cost comparisons are not what the efficiency score measures.

Question 49Methodology

A forecasting analyst applies simple exponential smoothing with smoothing constant 0.3 and obtains a forecast of 120 for the current period while the actual value turns out to be 130. What is the forecast for the next period?

  1. 123 Correct answer.
  2. 130
  3. 121
  4. 127

Explanation

The update is the current forecast plus the smoothing constant times the current forecast error, that is 120 plus 0.3 times 10, which equals 123. A larger smoothing constant would place more weight on the most recent observation.

Question 50Methodology

A time series exhibits a clear upward trend and a seasonal pattern with a period of four quarters. Which forecasting approach is most appropriate for producing multi-step forecasts?

  1. Holt-Winters exponential smoothing with additive trend and seasonal components Correct answer.
  2. Simple moving average with a window of four periods
  3. First-order exponential smoothing applied to the raw series
  4. A naive forecast that repeats the last observation

Explanation

Holt-Winters explicitly models level, trend, and seasonality, which matches a series with both a trend and a recurring seasonal pattern. Simpler methods capture at most one of these features and lag behind a trending series.