Prescriptive Analytics and Decision Science

Introduction and Context

Descriptive analytics explains what has happened, diagnostic analytics investigates why it happened, and predictive analytics estimates what is likely to happen. Prescriptive analytics addresses the subsequent managerial and engineering question: what should be done? It combines predictive information with objectives, constraints, uncertainty, available actions, and the consequences of alternative decisions.

Prescriptive analytics is therefore not merely an extension of forecasting. A forecast may predict demand of 10,000 units, but it does not determine how much to produce when capacity, labor, inventory, service commitments, and risk are considered. Prescriptive analytics uses mathematical optimization, simulation, scenario analysis, decision trees, and sensitivity analysis to recommend an action or policy.

Decision science provides the broader intellectual foundation. It integrates operations research, statistics, economics, behavioral decision theory, risk analysis, and data engineering. Its central concern is the disciplined transformation of evidence into defensible decisions under uncertainty.

In professional practice, a prescriptive model should not be treated as an automatic replacement for judgment. It is a structured decision aid. Its recommendation is useful only when the objectives, data, constraints, probability assumptions, and implementation conditions are credible.

Learning Objectives

After studying this chapter, students should be able to:

1. Distinguish predictive analytics from prescriptive analytics and decision science.

2. Formulate constrained decision problems using decision variables, objectives, and constraints.

3. Explain and apply linear, integer, nonlinear, and multi-objective optimization models.

4. Use simulation and scenario analysis to evaluate uncertain outcomes.

5. Construct and interpret decision trees using expected monetary value and risk measures.

6. Perform sensitivity and post-optimality analysis.

7. Combine optimization, simulation, and managerial judgment in practical decisions.

8. Evaluate ethical, governance, implementation, and model-risk issues.

1. The Architecture of a Prescriptive Decision

A prescriptive decision problem contains six fundamental elements:

• Decision variables: controllable choices, such as production quantities, staffing levels, routes, prices, or investment amounts.
• Objective function: the quantity to maximize or minimize, such as profit, service quality, cost, emissions, or risk.
• Constraints: restrictions imposed by resources, policy, technology, law, or contractual commitments.
• Parameters: values regarded as known for a particular model run, such as unit cost, capacity, or demand estimate.
• Uncertainty: unknown future conditions, including demand, failures, prices, and arrival rates.
• Decision context: the time horizon, decision-maker, stakeholders, and consequences of error.

A useful general representation is:

Choose x to maximize or minimize f(x, θ), subject to g(x, θ) ≤ 0 and h(x, θ) = 0,

where x is the vector of decisions and θ is the vector of parameters or uncertain inputs. The functions f, g, and h may be linear or nonlinear. This notation describes a model family rather than a single algorithm.

A strong formulation begins with the decision, not the available algorithm. Analysts should ask: Who makes the decision? Which actions are feasible? What outcome defines success? Which constraints are hard, and which can be violated at a penalty? What uncertainty matters to the decision?

2. Optimization Models

2.1 Linear programming

Linear programming (LP) is appropriate when the objective and constraints are linear functions of continuous decision variables. A general maximization model is:

Maximize Z = cᵀx

subject to Ax ≤ b, x ≥ 0.

Here, x is the vector of decision variables, c contains objective coefficients, A is the constraint-coefficient matrix, and b contains resource limits. All coefficients must be dimensionally consistent: for example, if x represents units produced, an objective coefficient may be monetary value per unit and a constraint coefficient may be labor hours per unit.

Illustrative formulation. A firm produces products A and B. Let x_A and x_B denote production quantities. Product A contributes $40 per unit and requires 2 machine hours and 1 labor hour. Product B contributes $30, requires 1 machine hour and 2 labor hours. Available resources are 100 machine hours and 80 labor hours.

Maximize Z = 40x_A + 30x_B

subject to:

2x_A + x_B ≤ 100 (machine hours)

x_A + 2x_B ≤ 80 (labor hours)

x_A, x_B ≥ 0.

At the intersection of the two resource constraints:

2x_A + x_B = 100 and x_A + 2x_B = 80.

Solving gives x_A = 40 and x_B = 20. The contribution is $2,200. The corner points include (0,0), (50,0), (0,40), and (40,20). Their contributions are $0, $2,000, $1,200, and $2,200 respectively. Thus, the intersection is optimal for this LP. Both resources are fully utilized.

The example assumes divisibility, constant unit contribution, known coefficients, and no interaction effects. If products must be produced in whole units, integer programming may be required.

2.2 Integer and mixed-integer programming

Integer programming requires some or all decision variables to be integers. Binary variables are especially useful for yes/no decisions: opening a facility, selecting a supplier, purchasing a technology, or assigning a project. Mixed-integer programming combines continuous and integer variables.

Integer restrictions make models more realistic but computationally harder. A binary variable y_j may represent whether facility j is opened. A linking constraint such as x_j ≤ My_j ensures that production x_j can occur only if y_j = 1, where M is a valid upper bound. An excessively large M can cause weak computational performance and numerical instability; it should be based on a justified physical or contractual limit.

Applications include vehicle routing, workforce scheduling, capital budgeting, network design, and portfolio selection. Solvers commonly use branch-and-bound or branch-and-cut methods, but the quality of the result still depends on the formulation and data.

2.3 Nonlinear and stochastic optimization

Nonlinear programming is used when returns, costs, physical relationships, or risk measures are nonlinear. Examples include economies of scale, congestion, portfolio variance, and energy-system behavior. Nonlinear models may have several local optima, so assumptions about convexity are important. In a convex minimization problem, any local optimum is global; without convexity, a solver’s result may depend on starting values and algorithmic settings.

Stochastic optimization explicitly represents uncertainty. In a two-stage model, some decisions are made before uncertainty is observed, while recourse decisions are made afterward. For example, a retailer chooses initial inventory before demand is known and then determines replenishment or allocation after demand information arrives. A basic objective may minimize expected total cost:

Minimize E[C(x, ξ)],

where x is the first-stage decision, ξ is a random vector, and C is the resulting cost. The expectation is meaningful only when the probability model for ξ is defensible.

2.4 Multi-objective optimization

Organizations rarely optimize a single objective. Profit may conflict with service level, environmental impact, resilience, or fairness. Common approaches include:

• Weighted-sum method: maximize w₁f₁(x) + w₂f₂(x), with weights reflecting preferences. Objectives should generally be normalized before weighting because units and scales differ.
• Epsilon-constraint method: optimize one objective while requiring other objectives to meet specified thresholds.
• Pareto analysis: identify solutions for which no objective can be improved without worsening another.

Weights are not purely technical parameters. They express values and should be documented, reviewed, and subjected to sensitivity analysis.

3. Simulation as a Decision Tool

Optimization selects among alternatives under a model. Simulation reproduces the behavior of a system under specified inputs and policies. It is especially useful when queues, failures, nonlinear interactions, or complex stochastic processes make closed-form analysis difficult.

3.1 Monte Carlo simulation

Monte Carlo simulation repeatedly samples uncertain inputs from specified probability distributions and calculates the resulting outcome. If Y is the performance measure and y₁, ..., y_N are simulated observations, the estimated mean is:

ȳ = (1/N) Σ y_i.

The sample standard deviation estimates variability. Under independent replications with finite variance, the standard error of the sample mean is s/√N, where s is the sample standard deviation. A simulation estimate should therefore report uncertainty, not only a point estimate.

Simulation results depend on the input distributions, correlations, random-number generation, warm-up treatment, and number of replications. In a queueing or service system, an initial warm-up period may be discarded to reduce bias from an artificial empty starting state. For comparing alternatives, common random numbers can reduce noise by exposing each alternative to comparable random conditions.

3.2 Discrete-event and agent-based simulation

Discrete-event simulation represents events such as arrivals, service completions, failures, and repairs. It is suitable for hospitals, call centers, manufacturing systems, and logistics networks. Agent-based simulation models individual entities and their rules, making it useful where adaptive behavior, heterogeneity, or social interaction matters.

Simulation does not automatically identify the best policy. Analysts must define candidate policies, performance measures, experimental design, and stopping criteria. Optimization and simulation can be combined through simulation optimization, in which an algorithm searches for a policy whose performance is evaluated by simulation.

4. Scenario Analysis and Robust Decisions

Scenario analysis evaluates decisions under coherent alternative futures rather than relying on a single forecast. A scenario is not merely a high, medium, or low number; it is a logically consistent set of assumptions. For example, a supply-chain scenario may combine high demand, port disruption, increased freight cost, and reduced supplier reliability.

A decision matrix lists alternatives and outcomes under scenarios. If probabilities are unavailable or disputed, robust methods may be preferable to expected-value optimization. Possible criteria include:

• Maximax: choose the alternative with the best possible outcome; optimistic and risk-seeking.
• Maximin: choose the alternative with the best worst-case outcome; conservative.
• Minimax regret: minimize the maximum loss relative to the best alternative in each scenario.
• Robust optimization: optimize against a specified uncertainty set rather than a precise probability distribution.

Scenario analysis is valuable for strategic planning, stress testing, cyber-risk planning, climate adaptation, and capital investment. Its limitations include scenario omission, internally inconsistent assumptions, and false confidence created by assigning arbitrary probabilities.

5. Decision Trees and Expected Value

Decision trees represent sequential decisions and uncertain events. Decision nodes are usually shown as squares, chance nodes as circles, and terminal nodes as outcomes. The tree must be read chronologically: decisions precede the uncertainty they can influence.

If outcomes have probabilities p_i and monetary consequences V_i, the expected monetary value (EMV) is:

EMV = Σ p_iV_i,

where probabilities are nonnegative and sum to one. EMV is appropriate when probabilities are credible and the decision-maker is approximately risk neutral over the relevant scale. It should not be interpreted as a guaranteed result.

Worked example. A company may launch a product immediately or conduct a market study costing $20,000. Without the study, demand is high with probability 0.6, producing a net payoff of $120,000; low demand has probability 0.4, producing $20,000. The immediate-launch EMV is:

0.6($120,000) + 0.4($20,000) = $80,000.

Suppose the study predicts favorable results with probability 0.55 and unfavorable results with probability 0.45. Given a favorable result, high demand probability is 0.85; given an unfavorable result, high demand probability is 0.30. The firm launches after a favorable result, but cancels after an unfavorable result. The favorable branch payoff before study cost is:

0.85($120,000) + 0.15($20,000) = $105,000.

The unfavorable branch chooses cancellation, with payoff $0. Therefore, study EMV is:

0.55($105,000) + 0.45($0) − $20,000 = $37,750.

Under these stated policies, immediate launch has the higher EMV. The study could still be justified for strategic, safety, or learning reasons, but not by this monetary calculation alone.

The value of sample information is the improvement in expected value from having information, before its cost. In practice, information may change actions only in some states; therefore, analysts should assess whether information is decision-relevant rather than merely accurate.

6. Risk, Utility, and Competing Objectives

Expected value can conceal dispersion and catastrophic downside. Two alternatives may have equal EMV but very different probabilities of loss. Risk analysis should examine variance, quantiles, probability of exceeding a loss threshold, worst-case outcomes, and downside measures such as expected shortfall.

A risk-neutral decision-maker evaluates monetary outcomes linearly. A risk-averse decision-maker may use a concave utility function U(V), choosing the alternative with the highest expected utility E[U(V)]. The utility function must be interpreted carefully: it represents preferences over consequences, not an objective probability model. In regulated or safety-critical contexts, constraints such as “probability of failure must not exceed 0.1%” may be more appropriate than a utility trade-off.

Risk registers should identify hazards, likelihood, impact, controls, owners, and residual risk. Correlation matters: simultaneous adverse events can make aggregate risk much greater than the sum suggested by independent assumptions.

7. Sensitivity and Post-Optimality Analysis

A recommendation is more credible when analysts determine which assumptions drive it. Sensitivity analysis changes one or more inputs and observes the effect on outcomes or decisions.

Important forms include:

• One-way sensitivity: vary one parameter while holding others constant.
• Two-way sensitivity: vary two parameters and display regions in which different decisions are optimal.
• Tornado analysis: rank variables by their influence on an outcome.
• Break-even analysis: identify the parameter value at which two alternatives have equal value.
• Scenario and stress testing: evaluate coordinated changes in assumptions.
• Post-optimality analysis: examine how objective coefficients or resource limits can change before the optimal basis or decision changes.

For the production LP above, compare the objective values of products at the intersection and relevant corner points. If the contribution of product A changes from $40 to 30, the intersection contribution becomes $30(40)+30(20)=$1,800, while producing only B yields $1,200 and only A yields $1,500; the intersection remains best among these points. Repeating such analysis over a range, rather than at one arbitrary value, reveals robustness.

Sensitivity analysis does not prove that a model is correct. It shows conditional stability: “If the model structure and other assumptions remain valid, this recommendation remains preferred over the tested range.”

8. Integrating Methods into a Decision Workflow

A practical prescriptive analytics workflow is:

1. Frame the decision. Specify the decision owner, timing, alternatives, objectives, and consequences.

2. Build a causal and operational understanding. Identify what actions can affect outcomes and what constraints are real.

3. Prepare data. Check definitions, missingness, outliers, units, time alignment, bias, and provenance.

4. Represent uncertainty. Use historical data, expert elicitation, statistical models, and stress scenarios. Document assumptions.

5. Formulate alternatives. Create a baseline or status-quo policy for comparison.

6. Optimize or simulate. Select a method matching the structure of the problem.

7. Validate. Test logic, compare with historical behavior, conduct extreme-value checks, and obtain domain review.

8. Perform sensitivity and risk analysis. Identify fragile assumptions and unacceptable outcomes.

9. Communicate the recommendation. State the action, rationale, assumptions, expected benefits, risks, and triggers for review.

10. Implement and monitor. Track realized outcomes, detect drift, and update the model.

Optimization and simulation are often complementary. Optimization may produce a schedule under estimated demand, while simulation tests that schedule under demand variability and disruptions. If simulation reveals poor robustness, the analyst can re-optimize using buffers, service constraints, or uncertainty sets.

9. Illustrative Case: Hospital Operating-Room Allocation

This is a fictional but realistic illustrative case.

Background

A regional hospital has six operating rooms and seeks to allocate weekly capacity among elective orthopedic, general, and oncology procedures. The hospital must improve contribution margin without violating clinical staffing limits or reducing urgent-case reserve.

Problem

Demand forecasts are uncertain. Orthopedic procedures have high contribution but require substantial specialist time. Oncology procedures have lower average contribution but high clinical priority. Cancellations create reputational and patient-care costs. The decision involves capacity allocation, staffing, and a reserve policy for urgent cases.

Analysis

The analytics team first defines procedure quantities as integer decision variables. Constraints include available room hours, surgeon hours, nursing hours, minimum oncology capacity, and a reserve of 10% of room capacity. A deterministic integer program produces a high-margin schedule. However, historical demand and cancellation data show that orthopedic demand is volatile and oncology demand is comparatively stable.

The team then constructs three coherent scenarios:

• Baseline: forecast demand and normal staffing.
• High orthopedic demand: increased orthopedic arrivals and specialist overtime pressure.
• Disruption: reduced nursing availability and elevated cancellation probability.

Each candidate schedule is evaluated through discrete-event simulation. Metrics include expected contribution, probability of exceeding staffing capacity, urgent-case fill rate, cancellation rate, and the fifth percentile of weekly contribution.

Approach and solution

The purely margin-maximizing schedule produces the highest average contribution but performs poorly in the disruption scenario. A second schedule reserves more room time and limits orthopedic allocation. Its average contribution is slightly lower, but it substantially improves urgent-case fill rate and reduces the probability of severe cancellation losses. A multi-objective formulation is used: contribution is maximized subject to minimum oncology capacity, urgent-case fill rate of at least 95%, and a bound on staffing overload probability.

The recommendation is therefore not simply “select the schedule with maximum expected margin.” It is a robust schedule accompanied by triggers: if nursing absence exceeds a defined threshold, elective orthopedic bookings are reduced; if urgent demand rises, reserve capacity is released according to priority rules.

Outcome

In the illustrative evaluation, the robust schedule has lower mean contribution than the unconstrained schedule but better downside performance and greater clinical reliability. The hospital adopts a pilot, monitors actual demand and cancellations, and reviews the policy monthly. Importantly, clinicians retain authority to override the schedule for patient-safety reasons, with overrides recorded for model improvement.

Learning points

1. A single objective can produce an operationally unsafe recommendation.

2. Forecast uncertainty must be connected to decisions and consequences.

3. Simulation can reveal risks hidden by average optimization results.

4. Clinical priorities should appear as constraints or governance rules, not informal afterthoughts.

5. A deployable policy includes monitoring and contingency triggers.

10. Implementation, Governance, and Ethical Issues

Prescriptive systems can amplify historical inequities. If an optimization model uses biased service data, it may allocate fewer resources to populations that were previously underserved. Fairness objectives and constraints should be explicitly considered, and outcomes should be audited across relevant groups.

Transparency is essential when decisions affect employment, healthcare, credit, education, or access to public services. Stakeholders should understand the decision logic, data sources, uncertainty, appeal procedures, and accountability structure. Human oversight must be meaningful rather than nominal.

Governance should include model ownership, version control, documentation, approval thresholds, access controls, monitoring, incident response, and retirement criteria. Data protection principles require purpose limitation, minimization, security, and lawful processing. In regulated environments, sector-specific rules may govern automated decisions, safety, financial risk, or medical operations.

AI and machine learning can improve forecasts, estimate demand distributions, discover patterns, and support simulation. However, a highly accurate prediction model does not guarantee a valid prescription. The causal effect of an action may differ from an observed association. Machine-learning recommendations should therefore be evaluated using experimentation where possible, causal analysis, counterfactual reasoning, and domain expertise. Generative AI may assist with model documentation or scenario generation, but numerical outputs require independent verification.

11. Challenges and Limitations

The principal challenges are often organizational rather than algorithmic:

• objectives are ambiguous or politically contested;
• constraints are omitted because they are difficult to quantify;
• data are stale, biased, incomplete, or measured after decisions;
• probability distributions are treated as facts rather than assumptions;
• models optimize proxies instead of real outcomes;
• implementation costs and behavioral responses are ignored;
• users overtrust precise numerical recommendations;
• feedback loops change the system after deployment.

There is also a distinction between model uncertainty and parameter uncertainty. Parameter uncertainty concerns the value of a quantity within a chosen model, such as demand variance. Model uncertainty concerns whether the chosen structure is appropriate at all, such as whether demand is independent across regions. Scenario analysis, alternative model classes, expert review, and stress testing are needed to address the latter.

Best Practices

1. Begin with a clearly owned decision and measurable outcomes.

2. Establish a transparent baseline policy.

3. Separate hard constraints from preferences and penalties.

4. Use units, definitions, and data lineage consistently.

5. Represent dependencies and correlations where material.

6. Report ranges, probabilities, and downside outcomes—not only expected values.

7. Compare optimization outputs with simulation and real-world feasibility.

8. Test sensitivity, stress, and distributional impacts.

9. Document assumptions, solver settings, overrides, and limitations.

10. Pilot decisions, monitor outcomes, and establish model-update triggers.

11. Preserve human accountability for high-impact decisions.

12. Treat robustness and explainability as design requirements.

Chapter Summary

Prescriptive analytics converts forecasts and evidence into recommended actions while accounting for objectives, constraints, uncertainty, and risk. Optimization models formalize feasible choices; linear, integer, nonlinear, stochastic, and multi-objective methods address different decision structures. Simulation evaluates dynamic and uncertain systems, while scenario analysis explores coherent alternative futures. Decision trees represent sequential choices and calculate expected value when probabilities and risk assumptions are appropriate. Sensitivity analysis identifies assumptions that influence recommendations and tests whether a decision is robust.

Effective decision science integrates these methods rather than treating any one as universally sufficient. A professional recommendation must be validated, communicated with its assumptions, monitored after implementation, and governed for fairness, privacy, safety, and accountability. The most valuable output is often not a single number, but a defensible policy with clear trade-offs, contingency triggers, and evidence about when the recommendation should change.

Key Terms

Prescriptive analytics: Analytics that recommends actions given objectives, constraints, and uncertainty.

Decision variable: A controllable quantity selected by a decision-maker or model.

Objective function: A mathematical representation of what the decision seeks to maximize or minimize.

Constraint: A requirement or limitation restricting feasible decisions.

Linear programming: Optimization with linear objective and constraint functions and continuous variables.

Integer programming: Optimization requiring one or more variables to take integer values.

Mixed-integer programming: Optimization combining continuous and integer variables.

Stochastic optimization: Optimization that explicitly represents uncertain parameters or outcomes.

Multi-objective optimization: Optimization involving two or more potentially conflicting objectives.

Pareto solution: A solution for which improving one objective requires worsening another.

Monte Carlo simulation: Repeated random sampling used to estimate uncertain outcomes.

Discrete-event simulation: Simulation in which system state changes at defined events.

Scenario analysis: Evaluation of alternatives under coherent possible futures.

Decision tree: A graphical representation of sequential decisions and uncertain events.

Expected monetary value: Probability-weighted average of monetary outcomes.

Expected utility: Probability-weighted value of outcomes transformed by a preference function.

Sensitivity analysis: Examination of how results change when assumptions or inputs vary.

Robust optimization: Optimization designed to perform acceptably across an uncertainty set.

Model risk: The possibility of harm caused by an incorrect, misused, or poorly implemented model.

Recourse: A later corrective decision made after uncertainty is revealed.

Review Questions

Short Answer Questions

1. How does prescriptive analytics differ from predictive analytics?

2. What are decision variables, objective functions, and constraints?

3. Why might simulation be used after optimization?

4. What is the purpose of sensitivity analysis?

5. Under what assumption is expected monetary value most appropriate?

Long Answer Questions

1. Explain the formulation and assumptions of a linear programming model, using a production example.

2. Compare integer programming, stochastic optimization, and multi-objective optimization.

3. Describe how a decision tree can support a sequential investment decision.

4. Explain how scenario analysis and robust optimization address uncertainty differently.

5. Discuss ethical, governance, and implementation issues in prescriptive analytics.

Analytical / Application Questions

1. A firm has two products, limited machine and labor capacity, and whole-unit production requirements. Formulate an integer programming model and identify the data needed to validate it.

2. Design a simulation experiment to compare two hospital staffing policies under uncertain arrivals and absenteeism. Specify inputs, outputs, replications, and validation checks.

3. An investment has uncertain returns and a possibility of catastrophic loss. Explain why EMV alone may be inadequate and propose a combined decision-tree, utility, and sensitivity-analysis approach.

Multiple Choice Questions with Answers

Question 1

Which element specifies what a prescriptive model seeks to maximize or minimize?

A. Decision variable

B. Objective function

C. Constraint

D. Scenario

Correct Answer: B. Objective function

Question 2

In the production example, the optimal continuous LP solution was to produce:

A. 40 units of A and 20 units of B

B. 50 units of A and 0 units of B

C. 0 units of A and 40 units of B

D. 20 units of A and 40 units of B

Correct Answer: A. 40 units of A and 20 units of B

Question 3

Which method is most appropriate when some decisions must be yes/no choices?

A. Linear programming with unrestricted continuous variables

B. Integer or mixed-integer programming

C. Descriptive statistics only

D. Moving-average forecasting only

Correct Answer: B. Integer or mixed-integer programming

Question 4

What is the expected monetary value of outcomes with probabilities p_i and values V_i?

A. The largest V_i

B. The unweighted sum of V_i

C. The probability-weighted sum of V_i

D. The variance of V_i

Correct Answer: C. The probability-weighted sum of V_i

Question 5

In the worked product-launch example, what was the EMV of launching immediately?

A. $20,000

B. $37,750

C. $80,000

D. $105,000

Correct Answer: C. $80,000

Question 6

Which technique evaluates a system by repeatedly sampling uncertain inputs?

A. Monte Carlo simulation

B. Integer rounding

C. Pareto dominance

D. Break-even accounting

Correct Answer: A. Monte Carlo simulation

Question 7

What does a Pareto solution represent?

A. A solution that violates no constraints but has zero value

B. A solution where no objective can improve without worsening another

C. The solution with the highest value on every objective

D. A random solution used for benchmarking

Correct Answer: B. A solution where no objective can improve without worsening another

Question 8

Which analysis changes one input while holding other inputs constant?

A. One-way sensitivity analysis

B. Agent-based simulation

C. Scenario omission

D. Integer relaxation

Correct Answer: A. One-way sensitivity analysis

Question 9

Why can expected monetary value be inadequate for safety-critical decisions?

A. It cannot use probabilities

B. It always selects the most expensive option

C. It may conceal dispersion and catastrophic downside

D. It applies only to integer variables

Correct Answer: C. It may conceal dispersion and catastrophic downside

Question 10

What is a key purpose of a baseline policy in prescriptive analytics?

A. To eliminate uncertainty from the data

B. To provide a reference for comparing proposed alternatives

C. To guarantee that the model is optimal

D. To replace validation and monitoring

Correct Answer: B. To provide a reference for comparing proposed alternatives