Prescriptive Analytics: From Data Analysis to Optimal Decisions in Economics and Management

1. What is prescriptive analytics?

Prescriptive analytics uses data and models to recommend actions that serve a specified objective under defined constraints. In management, the output might be a project portfolio, an order quantity, a staff schedule or a resource-allocation plan.

Delen (2019) presents methods including optimization, simulation and multicriteria decision-making. Bertsimas and Kallus (2020) develop an approach that connects data with decision models to turn predictive information into operational choices. This guide starts with a problem whose parameters are given, allowing readers to understand the decision structure before extending it to machine learning and uncertainty.

Shmueli (2010) distinguishes explanation from prediction, with implications for model development and evaluation. Prescriptive analytics also requires specifying actions, their costs and their limits. An accurate predictive model must therefore be connected to a particular decision problem before it can produce a useful recommendation.

2. Framing a decision problem

Define the following five components before choosing software. This prevents collecting data and tools without knowing which decision the analysis is intended to support.

• Decision variables: what the manager can choose, such as whether to select a project, how much to produce or how many hours to allocate to an activity.
• Objective function: how the desired outcome is quantified, for example maximizing emission reductions or minimizing the cost of meeting demand. Specify units and the evaluation period.
• Constraints: mandatory conditions concerning budgets, staffing, capacity, time, technical requirements and project dependencies.
• Input parameters: costs, effectiveness, demand and capacity. Record the source, date and degree of certainty of each parameter.
• Recommendation criteria: how the recommendation is compared with current practice and checked when conditions change.

Project-selection decisions are often coded 0 or 1, whereas material-allocation variables may be continuous. The variable type must reflect implementation: a packaged equipment system cannot be modeled as an investment that can be divided arbitrarily.

3. Matching methods to the question

These methods can complement one another. Simulation can assess variation in a portfolio's effectiveness, while optimization seeks a portfolio meeting the selection criteria. With multiple objectives, make trade-offs explicit: money and tonnes of CO₂e cannot be added directly without an appropriate conversion or normalization rule.

4. Case: selecting green projects with a VND 300 million budget

A firm is considering four energy-saving and emission-reduction projects. Initial investment must not exceed VND 300 million. The objective is to maximize expected emission reductions over one full operating year. Each project is selected in full or not selected at all.

4.1. Baseline model assumptions

• Projects can be implemented independently; none must be selected before another.
• Emission reductions are additive, after removing benefits counted twice across projects.
• Investment costs and expected effectiveness are treated as fixed in the baseline solution.
• All projects use the same full operating year and emissions-accounting boundary.
• The current objective is emission reduction; operating costs, financial benefits, useful lives and terminal values are not modeled.

Additivity requires careful checking in practice. Electricity savings, for example, change the demand a solar system must meet. Counting the same electricity benefit for both projects overstates the total reduction. Where projects interact, adjust the parameters or model benefits for combinations of projects.

4.2. Replacing hypothetical values with company data

Collect quotations, energy-consumption data, operating hours, baselines and emission-reduction estimates. Record each parameter's source, date and accounting boundary, and check whether conversion factors fit the application. Parameters with the greatest influence on the recommendation deserve the closest validation.

5. Building the optimization model

Let x_i = 1 if project i is selected and x_i = 0 otherwise, for i in {A, B, C, D}. The baseline model is:

The objective Z is measured in tonnes CO₂e/year, and the left side of the budget constraint is in VND million. This is a linear optimization model with binary variables: every expression sums decision variables multiplied by fixed coefficients.

A feasible solution satisfies every constraint. An optimal solution has the best objective value within the feasible set. Here, A + B is feasible, whereas C + D achieves a higher objective and is optimal for the baseline model.

6. Working in Excel Solver

Microsoft describes Solver as finding a maximum or minimum for an objective cell by changing decision cells subject to constraints. Its project-selection guidance uses binary variables and SUMPRODUCT, matching this exercise's structure (Microsoft, n.d.-a, n.d.-b).

6.1. Organizing data and formulas

Enter headings in row 1 and projects A, B, C and D in rows 2–5. Set up the following ranges:

Depending on regional settings, Excel may require semicolons instead of commas in formulas. Costs and effectiveness must be numeric; put units in headings rather than mixing numbers and words in calculation cells.

6.2. Setting up and solving the model

1. In desktop Excel for Windows, enable Solver through File → Options → Add-ins → Manage: Excel Add-ins → Go, then select Solver Add-in. Open Data → Solver.
2. Set Objective to $D$7 and select Max.
3. Set By Changing Variable Cells to $E$2:$E$5.
4. Add $C$7 <= $C$9 and a bin constraint for $E$2:$E$5.
5. Choose Simplex LP because the objective and constraints are linear; confirm that Ignore Integer Constraints is not selected.
6. Choose Solve, read the termination message and retain the solution with Keep Solver Solution.

Check E2:E5 = 0, 0, 1, 1; C7 = 300; D7 = 125. Dialog labels may vary slightly with the software version and installed language.

6.3. Checking termination and optimality

For integer problems, a message that a solution was found within tolerance must be interpreted using Integer Optimality %. This concerns the objective gap to the best known bound, distinct from constraint-feasibility precision. Frontline Systems explains that positive tolerance can allow termination at a near-optimal solution (Frontline Systems, n.d.-a, n.d.-b).

For this four-variable exercise, Integer Optimality % can be set to 0 and all 16 combinations checked. Larger problems require suitable tolerances and time limits, with the solver status reported accurately. A solution obtained at a time limit is not automatically a proven optimum.

7. Checking the result in R: all 16 combinations

Four binary (0–1) variables produce 2⁴ = 16 combinations. R's expand.grid() creates combinations of the supplied values (R Core Team, n.d.). The code below uses base R to calculate costs and effectiveness, filter feasible portfolios and sort by the objective.

du_an <- data.frame(
  ma = c("A", "B", "C", "D"),
  chi_phi = c(90, 140, 200, 100),
  giam = c(30, 55, 80, 45)
)

to_hop <- expand.grid(
  A = 0:1, B = 0:1, C = 0:1, D = 0:1,
  KEEP.OUT.ATTRS = FALSE
)

giai <- function(ngan_sach = 300, giam = du_an$giam) {
  X <- as.matrix(to_hop[, du_an$ma, drop = FALSE])
  kq <- to_hop
  kq$chi_phi <- as.vector(X %*% du_an$chi_phi)
  kq$giam_phat_thai <- as.vector(X %*% giam)
  hop_le <- kq[kq$chi_phi <= ngan_sach, , drop = FALSE]
  hop_le[order(-hop_le$giam_phat_thai, hop_le$chi_phi),
          , drop = FALSE]
}

# Top option: A=0, B=0, C=1, D=1; cost=300; reduction=125
giai(300)[1, ]

# Budget levels: 250, 300, 350, 400
lapply(c(250, 300, 350, 400), function(b) giai(b)[1, ])

# Scenario C reduces only 30 tonnes CO₂e/year
giai(300, giam = c(30, 55, 30, 45))[1, ]

When two portfolios have equal emission reductions, the code prefers the cheaper one. This is an additional sorting rule; the baseline model maximizes only emission reduction, so equal-objective solutions may both be optimal.

Enumeration suits small problems. The number of combinations grows as 2ⁿ; larger numbers of projects and constraints require an appropriate optimization solver rather than mechanically extending enumeration.

8. Interpreting and reporting the results

In the baseline model, C + D is optimal: VND 300 million investment and an expected reduction of 125 tonnes CO₂e/year.

Among two-project portfolios, A + C yields 110 tonnes, B + D 100 tonnes, A + B 85 tonnes and A + D 75 tonnes. B + C yields 135 tonnes but costs VND 340 million, exceeding the budget. Every portfolio with at least three projects costs more than VND 300 million; each individual project yields at most 80 tonnes.

8.1. Comparing with current practice

Suppose the firm initially intended to choose A + B. Moving to C + D raises expected reductions from 85 to 125 tonnes CO₂e/year while increasing investment from VND 230 to 300 million. The differences are 40 tonnes CO₂e/year and VND 70 million in initial investment.

This comparison uses the exercise's environmental objective. A financial assessment also requires cash flows, operating costs, a time horizon and appropriate economic criteria. Without total baseline emissions, it is not possible to infer the percentage reduction in the firm's overall emissions.

8.2. Sample reporting language

With an initial investment budget capped at VND 300 million, the binary project-selection model identified projects C and D as the portfolio with the greatest expected emission reduction: 125 tonnes CO₂e over one full operating year. The result was checked against all 16 combinations. The conclusion assumes additive project benefits, fixed costs and no project dependencies. Costs and effectiveness should be validated before implementation, particularly the emission reduction expected from project C.

This wording applies to the hypothetical example. An empirical study must substitute actual values, describe their sources and report its own verification conditions.

9. Sensitivity analysis: how does the recommendation change?

9.1. Changing the budget and solving again

Keep baseline effectiveness parameters fixed, change the budget in C9 and solve again. The following results were verified by enumerating all combinations at each budget.

A larger budget does not require spending every available unit. At VND 350 million, B + C spends VND 340 million. The budget-outcome relationship is stepwise because projects are indivisible. For integer models, sensitivity analysis should vary parameters and re-solve; use caution when interpreting concepts developed for continuous linear programming.

9.2. Why is ranking benefit-to-cost ratios insufficient?

Emission reductions per VND million are approximately 0.333 for A, 0.393 for B, 0.400 for C and 0.450 for D. With a VND 350 million budget, ranking projects by these ratios selects D then C, spends VND 300 million and yields 125 tonnes. B + C instead spends VND 340 million and yields 135 tonnes.

Ranking individual projects does not guarantee the best combined portfolio. Indivisibility and the budget constraint make evaluating combinations necessary.

9.3. Changing project dependencies

If selecting C requires B, any portfolio containing C costs at least VND 340 million and is infeasible under a VND 300 million budget. Re-solving selects B + D, spending VND 240 million and yielding 100 tonnes CO₂e/year. Omitting one real constraint can change the entire recommendation.

10. Decisions under uncertainty

10.1. Examining an adverse scenario

Suppose C's effectiveness is uncertain. The exercise defines two states: a baseline in which C reduces emissions by 80 tonnes CO₂e/year, and an adverse state in which it reduces only 30 tonnes. A, B and D retain their effectiveness, and the budget stays at VND 300 million.

In the adverse state, C + D reduces only 75 tonnes CO₂e/year; B + D yields 100 tonnes and would be optimal if that state were known in advance. Re-solving individual scenarios reveals sensitivity, but the manager must still choose a portfolio before knowing which state will occur.

10.2. Selecting a portfolio conservatively

One option is to maximize the minimum outcome across the defined scenarios, or max–min. Bertsimas and Sim (2004) discuss the trade-off between nominal performance and protection against uncertain inputs. This two-scenario example illustrates that conservative principle simply; it does not implement their particular uncertainty-budget model.

Checking every feasible option shows that B + D maximizes the minimum outcome across the two states, at 100 tonnes CO₂e/year. It costs VND 240 million. Relative to C + D, the firm sacrifices some baseline performance to improve performance in the adverse state.

Extend the model by introducing z and maximizing it while retaining the budget and binary constraints:

These constraints prevent z from exceeding the portfolio's outcome in either state. Maximizing z finds the portfolio with the best worst-case outcome. Protection applies only to the chosen scenario set; omitted scenarios may still change the result.

10.3. Which information deserves validation first?

Holding everything else constant, C + D outperforms B + D when C reduces emissions by more than 55 tonnes CO₂e/year. At 55 tonnes, both portfolios yield 100 tonnes, but B + D is cheaper. This threshold focuses validation on C's effectiveness rather than relying on an unchecked point estimate.

11. Connecting with SEM and PLS-SEM in management research

In economics and management research, SEM supports measurement assessment and the study of hypothesized relationships. Resource allocation additionally requires defining implementable actions and how they change outcomes. That requires further research design and data.

A standardized path coefficient is not the return on an investment. For example, β = 0.40 between green capability and environmental performance does not mean an additional VND one million in training improves environmental performance by 0.40 units. These quantities have different meanings and units.

Budget recommendations require evidence connecting expenditure, implementation and outcomes, together with identification assumptions when interpreting interventions causally. Pearl (2010) explains the role of assumptions about data-generating mechanisms in intervention inference. Observational survey data do not settle that problem merely because coefficients are statistically significant. Shmueli's (2010) explanation-prediction distinction also helps avoid conclusions that exceed a model's purpose and evidence.

A study may combine these components when the question requires them. Each needs a clear role and explicit transition conditions. Additional techniques add value only when they provide information needed for the decision.

12. Common errors and diagnostic checks

Deleting a constraint simply to make an infeasible model run can produce an impossible recommendation. Distinguish mandatory conditions from negotiable objectives, then revise the problem with the decision-maker.

13. Workflow for an applied study

Checklist before making a recommendation

• The objective, units and evaluation period are agreed.
• Variable types reflect whether projects are divisible.
• Costs, effectiveness and baselines have verifiable sources.
• Dependencies, exclusions, double-counting and interactions have been checked.
• The objective and resource use have been independently checked.
• Solver status and stopping conditions are recorded accurately.
• Comparison with current practice uses the same criteria.
• Recommendations state their conditions and monitoring plan.

For data-driven predictive models, evaluate out-of-sample situations using the actual decision criterion, such as cost or demand fulfillment. Information available only after the decision can make a policy's apparent performance better than what is achievable in practice.

15. Further reading

Prescriptive Analytics: Dursun Delen

Pearson FT Press released this book in 2019, with copyright ©2020. It introduces optimization, simulation, multicriteria decision-making and decision-support systems.

Use it to extend the project-selection example to other types of problems. This guide's description of the book was checked against the publisher's overview and contents.

This is an independent applied guide. Its case, data and scenarios were constructed for teaching and are not results from a particular firm. Numerical results were checked by enumerating every combination.