Prescriptive Analytics and Optimization

Introduction

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 next managerial question: what should be done? It combines forecasts, business rules, mathematical models, and computational methods to recommend actions that best satisfy organizational objectives.

Optimization is a central technique in prescriptive analytics. An optimization model represents a decision problem through:

• Decision variables: controllable choices, such as quantities to produce, prices to charge, or vehicles to assign.
• An objective function: a mathematical expression to maximize or minimize.
• Constraints: limitations or requirements involving resources, policies, demand, capacity, or legal rules.
• Parameters: known or estimated inputs, such as costs, revenues, capacities, and probabilities.

A sound optimization model is not merely a mathematical exercise. It is a structured representation of a real decision, and its usefulness depends on the quality of its assumptions, data, implementation, and managerial interpretation.

Learning Objectives

By the end of this chapter, you should be able to:

1. Translate business decisions into decision variables, objective functions, and constraints.
2. Formulate and interpret linear programming models.
3. Explain graphical, simplex-based, and solver-based approaches to linear optimization.
4. Distinguish continuous, integer, binary, network, and multi-objective models.
5. Apply optimization to pricing, allocation, scheduling, logistics, and supply-chain decisions.
6. Interpret shadow prices, reduced costs, sensitivity information, and trade-offs.
7. Evaluate optimization models critically, including data, ethical, governance, and implementation risks.

1. The Logic of Prescriptive Analytics

Prescriptive analytics begins with a decision context. A business may need to determine how much of each product to make, which customers to serve, where to locate facilities, how to schedule employees, or how to route deliveries. Forecasts may provide demand estimates, but forecasts alone do not determine the best action.

The general optimization structure is:

Here, x is the vector of decision variables, f(x) is the objective function, g_i(x) represents constraint functions, b_i contains resource or policy limits, and X specifies allowable values such as nonnegativity or integrality. Linear programming is obtained when the objective and constraints are linear functions.

A feasible solution satisfies every constraint. The feasible region is the set of all feasible solutions. An optimal solution is a feasible solution that gives the best objective value according to the chosen maximization or minimization direction. A problem may have one optimal solution, multiple optimal solutions, or no feasible solution. In some cases, the objective can improve without bound.

Model-building sequence

A practical modeling sequence is:

1. Define the decision and the planning horizon.
2. Identify the decision variables and their units.
3. Specify the objective and whether it is to be maximized or minimized.
4. Translate operational, financial, legal, and service requirements into constraints.
5. State assumptions and data sources.
6. Check units, signs, bounds, and logical consistency.
7. Solve and validate the model.
8. Interpret the recommendation and test its sensitivity.

The distinction between a mathematical optimum and a managerial decision is important. An optimum may be infeasible in practice because of labor relations, supplier reliability, implementation time, or omitted constraints.

2. Linear Programming Formulation

A linear programming (LP) model assumes proportionality, additivity, divisibility, and known coefficients over the modeled range. Its standard maximization form is:

The symbol x_j is the quantity of activity or product j; c_j is its contribution to profit or value per unit; a_ij is the amount of resource i used per unit of j; and b_i is the available amount of resource i. The units must be consistent. For example, if a_ij is labor hours per unit and x_j is units, the left side is labor hours and b_i must also be labor hours.

A minimization model may represent cost, distance, emissions, or delivery time. Constraint directions can be ≤, =, or ≥. Upper and lower bounds can be written as:

where L_j and U_j are lower and upper bounds.

Example: product-mix decision

A manufacturer produces Standard and Premium units. Let:

• S = Standard units produced per week;
• P = Premium units produced per week;
• contribution margins be £30 and £50 per unit, respectively;
• each Standard unit require 2 machine hours and 1 labor hour;
• each Premium unit require 4 machine hours and 3 labor hours;
• weekly capacities be 400 machine hours and 240 labor hours.

The model is:

The feasible corner points include (0, 0), (200, 0), (0, 80), and the intersection of the two resource constraints. Solving the intersection:

Multiplying the second equation by 2 and subtracting the first gives 2P = 80, so P = 40 and S = 120. Objective values are:

• (200, 0): Z = 30(200) = 6000;
• (0, 80): Z = 50(80) = 4000;
• (120, 40): Z = 30(120) + 50(40) = 5600.

Thus, under the stated assumptions, producing 200 Standard units and no Premium units yields the highest contribution, £6,000. This result may initially seem surprising because Premium has a higher unit margin. Standard uses fewer machine and labor hours, and the relative resource requirements determine the best mix.

The model assumes divisibility, constant margins, known capacities, and no additional demand or production constraints. If customers demand at least some Premium units, or if units must be produced in batches, the model must be changed.

3. Solving and Interpreting Linear Models

For two variables, the graphical method plots each constraint, identifies the feasible region, and evaluates the objective at extreme points. The fundamental result is that, when an LP has an optimal solution and the feasible region is nonempty and bounded in the relevant direction, at least one optimum occurs at an extreme point of the feasible region.

For larger models, software uses algorithms such as the simplex method, revised simplex, or interior-point methods. A solver does not replace modeling judgment: it searches the mathematical model supplied by the analyst.

Duality and shadow prices

Every appropriately formulated LP has a related dual problem. The dual gives an economic interpretation of resource constraints. The shadow price of a binding resource is the marginal change in the optimal objective value resulting from a one-unit increase in that resource, within the valid sensitivity range.

If a machine-hour constraint has a shadow price of £8, one additional machine hour is worth approximately £8 in improved optimal contribution, provided other assumptions remain valid and the change is small enough. A nonbinding resource generally has a shadow price of zero because extra units are not currently valuable.

Shadow prices are not automatically market prices. They are conditional values based on the model, current data, and allowable range. A large shadow price may indicate a bottleneck, but purchasing more capacity may be impractical or may invalidate the linear assumptions.

Reduced costs and sensitivity

A reduced cost indicates how much the objective coefficient of a currently zero variable would need to improve, under standard LP conventions, before that variable becomes attractive in an optimal basis. Sensitivity analysis examines how changes in costs, revenues, capacities, demand, and processing times affect the recommendation.

Sensitivity analysis should be complemented by scenario analysis. A scenario changes several inputs coherently—for example, a recession may reduce demand and prices while increasing payment risk. A model that is optimal in one scenario may be fragile across plausible alternatives.

4. Integer and Binary Optimization

LP permits fractional decisions, but many business decisions are indivisible. Integer programming restricts some or all variables to integer values:

Binary variables take only two values:

They are useful for yes/no decisions such as opening a warehouse, selecting a supplier, assigning an employee to a shift, or launching a promotion.

A fixed-charge facility model can be written as:

Here, F_k is the fixed cost of opening facility k, c_ik is the variable cost of serving demand point i from facility k, d_i is demand, and M_ik is a valid upper bound. The linking constraint prevents shipments from a facility that is not opened. An excessively large M_ik can weaken computational performance; bounds should be as tight as logically justified.

Mixed-integer programming (MIP) combines continuous, integer, and binary variables. Common solution approaches include branch-and-bound and branch-and-cut. Solvers may stop with a proven optimality gap when exact optimality is computationally expensive. If the incumbent solution has objective value V and the best bound is B, a relative gap can be expressed, subject to the solver's convention, as:

The analyst must verify whether the solver reports a minimization or maximization gap and whether the solution is acceptable operationally.

Logical constraints

Binary variables can encode business logic. If project A requires project B, then y_A ≤ y_B. If exactly one of two alternatives must be selected, then y₁ + y₂ = 1. If at most one may be selected, then y₁ + y₂ ≤ 1. Such statements should be tested with sample combinations before solving.

5. Network Optimization

Network models represent nodes and arcs. Nodes may be warehouses, stores, airports, or time periods; arcs may represent transportation links, production movements, or feasible assignments.

In a minimum-cost flow model, let x_ij be flow from node i to node j, and c_ij its unit cost. A flow-conservation constraint is:

where s_i is external supply at node i and d_i is external demand. Depending on the formulation, supply and demand signs may be represented through a single net-balance parameter. Arc capacities can be imposed as 0 ≤ x_ij ≤ u_ij.

Important network problems include:

• Shortest path: minimize travel distance or time between an origin and destination.
• Maximum flow: maximize quantity through a network subject to arc capacities.
• Transportation: allocate supply from origins to demand destinations at minimum cost.
• Assignment: match agents to tasks, usually with binary variables.
• Minimum spanning tree: connect all nodes with minimum total edge cost and no cycles.

Network structure often permits specialized algorithms that are faster and easier to interpret than general-purpose MIP. However, real applications may require integer fleet sizes, time windows, vehicle capacities, multiple products, or uncertain travel times, making hybrid models necessary.

6. Scheduling and Allocation Applications

Scheduling models assign activities to people, machines, rooms, or time periods. A binary variable may be defined as y_it = 1 if employee i works shift t, and 0 otherwise. Coverage constraints ensure sufficient staffing:

where r_t is the required number of employees in shift t. Availability, maximum hours, rest periods, qualifications, and fairness constraints are added as appropriate.

Allocation models distribute scarce resources among competing uses. For example, advertising budget allocation might maximize expected conversions:

This linear form assumes constant marginal value. If advertising response diminishes with spending, piecewise-linear approximations, nonlinear programming, or discrete campaign choices may be more suitable.

7. Pricing and Revenue Optimization

Pricing decisions often combine demand forecasts with optimization. If demand for product j is estimated as:

where p_j is price, q_j is expected demand, and a_j, b_j are estimated parameters, expected contribution is:

where v_j is unit variable cost. This model assumes a valid linear demand approximation over the relevant price range, stable competitive conditions, and no stockout or capacity effects. In practice, prices may be restricted to approved values, customer segments may have different willingness to pay, and fairness or competition rules may constrain price differentiation.

Pricing optimization must not be used to coordinate with competitors, exploit legally protected groups, or conceal discriminatory outcomes. Governance should include approval thresholds, monitoring, audit trails, and tests for disparate impact.

8. Multi-Objective Optimization

Business decisions commonly involve competing goals: profit, service quality, emissions, resilience, and employee welfare. There may be no single solution that simultaneously optimizes every objective.

A weighted-sum model is:

where f_k(x) are objectives converted to comparable, preferably dimensionless scales, and w_k ≥ 0 are weights. Usually, weights sum to 1, although normalization is a modeling convention rather than a mathematical necessity.

Because profit, emissions, and service may have different units and directions, raw values should not be added without normalization. A common benefit scaling is:

where the denominator is positive and the minimum and maximum are defined reference values. For a cost objective to be minimized, the scale can be reversed. Reference values must be documented because weights have meaning only relative to the scaling.

Another approach is the epsilon-constraint method: optimize one primary objective while imposing limits on others, such as minimizing cost subject to emissions ≤ a target and service level ≥ a target. The set of nondominated solutions is called the Pareto frontier. A solution is Pareto-efficient if improving one objective would worsen at least one other objective.

Multi-objective analysis makes value judgments visible. It does not eliminate them. Decision makers should review weights, targets, and stakeholder consequences rather than treating the model's output as value-neutral.

9. An Illustrative Supply-Chain Case

Background

The following is a fictional illustrative case. Northstar Foods supplies three regional markets from two plants. Management must decide weekly production and shipments while minimizing cost and maintaining service.

Problem

Plant A can produce up to 600 cases and Plant B up to 500 cases. Market 1 requires 400 cases and Market 2 requires 500 cases. Production costs are £4 per case at A and £5 per case at B. Shipping costs per case are:

Let x₁1, x₁2, x₂1, and x₂2 denote shipments from Plants A and B to Markets 1 and 2. The model is:

The coefficients combine production and shipping cost. For example, Plant A to Market 1 costs £4 + £1 = £5 per case.

Analysis and solution

Plant A is cheaper for Market 1 by £2 per case relative to Plant B, whereas Plant B is cheaper for Market 2 by £1 per case relative to Plant A. Therefore, allocate Plant A first to Market 1, subject to its capacity. Supplying all 400 cases of Market 1 from A leaves 200 units of A capacity. Plant B supplies 500 cases to Market 2, using its full capacity, and the remaining 200 cases of Market 2 come from A.

Thus:

• x₁1 = 400;
• x₁2 = 200;
• x₂1 = 0;
• x₂2 = 300.

The total cost is:

Both markets receive their requirements, Plant A uses 600 cases, and Plant B uses 300 of its 500-case capacity. The result is optimal for this transportation structure because the relative savings support assigning Plant A to Market 1 and Plant B to Market 2 before using the less favorable routes.

Managerial interpretation

The model identifies Plant A capacity as strategically valuable because it is particularly economical for Market 1. However, the recommendation depends on reliable costs and demand. If Market 2 demand rises above 500, Plant A must cover more of it. If Plant B becomes unavailable, the model may remain feasible only if total capacity and route restrictions permit service. A robust extension could reserve emergency capacity or penalize unmet demand.

Learning points

The case illustrates four principles: formulate flows explicitly, combine cost components consistently, verify every balance and capacity constraint, and examine scenarios rather than relying on one deterministic optimum.

10. Implementation, Governance, and Technology

Optimization projects typically use spreadsheets, SQL databases, Python or R modeling libraries, and commercial or open-source solvers. A reliable implementation separates data preparation, model generation, solving, validation, and reporting. Version control should preserve data snapshots, model code, parameter settings, solver versions, and approvals.

AI and machine learning can improve optimization by providing demand forecasts, travel-time estimates, customer-response models, and anomaly detection. However, predictive accuracy does not guarantee a good prescription. Forecast bias can systematically direct inventory away from underserved areas; automated prices can create unfair outcomes; and opaque recommendations can undermine accountability.

Good governance includes:

• documented assumptions and data lineage;
• validation against historical and operational cases;
• scenario and stress testing;
• human review for high-impact decisions;
• access controls and audit logs;
• monitoring for drift, bias, constraint violations, and unexpected outcomes;
• a clear override and escalation process.

Optimization models should not encode illegal discrimination, unsafe working conditions, deceptive pricing, or impossible service promises. Constraints can express ethical requirements, such as minimum service for vulnerable communities, maximum working hours, or emissions limits. These are not merely technical additions; they represent organizational values and obligations.

11. Challenges and Limitations

Data and parameter uncertainty

Costs, demand, capacities, and travel times are estimates. If uncertainty is material, analysts may use scenario analysis, stochastic programming, robust optimization, or rolling-horizon reoptimization. These methods do not remove uncertainty; they make the response to uncertainty more explicit.

Model misspecification

A mathematically precise model can still be wrong if it omits setup times, quality constraints, minimum order quantities, labor rules, or customer behavior. Stakeholder interviews and pilot testing are essential.

Computational complexity

Integer and nonlinear models can be difficult to solve at scale. Decomposition, heuristics, valid inequalities, aggregation, and carefully chosen bounds may improve performance. A near-optimal solution delivered quickly may be more useful than an exact solution delivered after the decision window has closed.

Organizational adoption

Users may reject an optimization recommendation if they cannot understand it or if it conflicts with tacit knowledge. Explainability should include the objective, binding constraints, key trade-offs, sensitivity, and reasons for major assignments. Adoption improves when users can test scenarios and provide structured feedback.

Feasibility and resilience

A cost-minimizing plan may have no buffer for disruptions. Resilience may require redundant suppliers, safety stock, flexible capacity, or explicit service penalties. The cheapest expected plan is not always the best plan when disruption costs are severe.

Best-Practice Checklist

Before releasing an optimization recommendation, the analyst should ask:

1. Is the decision clearly defined, including horizon and authority?
2. Are all variables, units, bounds, and coefficients documented?
3. Does every constraint represent a real requirement or justified policy?
4. Is the objective aligned with organizational and stakeholder value?
5. Has feasibility been independently checked?
6. Have sensitivity, scenario, and stress tests been performed?
7. Are integer and logical requirements correctly modeled?
8. Have legal, ethical, fairness, privacy, and safety implications been reviewed?
9. Can decision makers understand and challenge the recommendation?
10. Is there a monitoring and reoptimization process after implementation?

Chapter Summary

Prescriptive analytics converts business information into recommended action. Optimization models formalize decisions through variables, objectives, constraints, and parameters. Linear programming is appropriate when relationships are linear and decisions can be continuous; integer and binary models represent indivisible choices and logical conditions. Network models support transportation, assignment, routing, and flow decisions. Scheduling and allocation models distribute scarce resources under operational rules.

Shadow prices and reduced costs provide valuable economic interpretation, while sensitivity and scenario analysis reveal how recommendations respond to changing assumptions. Pricing optimization links demand models to contribution, but requires careful attention to capacity, competition, fairness, and regulation. Multi-objective methods make trade-offs among profit, service, sustainability, and resilience explicit through weighted objectives, epsilon constraints, or Pareto analysis.

A solver can identify the best solution to a stated model, but it cannot decide whether the model is complete, ethical, or strategically appropriate. Effective prescriptive analytics therefore combines mathematical rigor, data quality, stakeholder judgment, governance, and ongoing monitoring.

Key Terms

Decision variable: A controllable quantity or choice represented in an optimization model.

Objective function: A mathematical expression to maximize or minimize.

Constraint: A mathematical representation of a limitation, requirement, or policy.

Parameter: An input value treated as known or estimated in a model.

Feasible solution: A decision that satisfies every model constraint.

Feasible region: The set of all feasible solutions.

Linear programming: Optimization with linear objective and constraint functions.

Integer programming: Optimization in which specified variables must take integer values.

Binary variable: A variable restricted to 0 or 1.

Mixed-integer programming: A model containing continuous and integer or binary variables.

Shadow price: The marginal objective value of relaxing a constraint within a valid range.

Reduced cost: The objective-coefficient improvement required for a currently inactive variable to become attractive under LP conventions.

Network optimization: Optimization using nodes and arcs to represent flows, connections, or assignments.

Transportation model: A network model allocating supply to demand at minimum cost.

Assignment model: A model matching agents or resources to tasks.

Pareto frontier: The set of nondominated solutions in a multi-objective problem.

Sensitivity analysis: Examination of the effect of input changes on a solution.

Scenario analysis: Evaluation of coherent alternative assumptions about the operating environment.

Optimality gap: A measure of the difference between an incumbent solution and a bound on the best possible solution.

Prescriptive analytics: Analytics that recommends actions under objectives and constraints.

Review Questions

5 Short Answer Questions

1. What are the three fundamental components of an optimization model?
2. Why must units be consistent in a linear programming model?
3. What is the difference between an integer variable and a binary variable?
4. What does a shadow price represent?
5. Why is normalization often needed in a weighted multi-objective model?

5 Long Answer Questions

1. Explain the process of translating a supply-chain decision into decision variables, an objective function, and constraints. Include a small illustrative formulation.
2. Compare linear programming, mixed-integer programming, and network optimization in terms of assumptions and business applications.
3. Explain how sensitivity analysis, shadow prices, and scenario analysis support managerial decision making.
4. Discuss the benefits and limitations of using optimization for pricing and revenue decisions.
5. Explain how ethical, legal, and governance requirements can be incorporated into prescriptive analytics.

3 Analytical / Application Questions

1. A firm has two products, limited labor and machine capacity, and different contribution margins. Formulate a product-mix LP, identify assumptions, and explain how you would validate the result.
2. Design a binary optimization model for selecting warehouses when opening costs, transportation costs, capacity limits, and customer-service requirements are present.
3. A logistics manager must balance cost, delivery reliability, and emissions. Propose a multi-objective formulation, explain how objectives could be normalized, and describe how management could select a Pareto-efficient solution.

Multiple Choice Questions with Answers

Question 1

Which element of an optimization model represents a controllable business choice?

A. Decision variable

B. Parameter

C. Shadow price

D. Feasible region

Correct Answer: A. Decision variable

Question 2

In the product-mix example, what is the contribution from producing 200 Standard units and no Premium units?

A. £4,000

B. £5,000

C. £6,000

D. £8,000

Correct Answer: C. £6,000

Question 3

Which variable type is most appropriate for a warehouse opening decision?

A. Continuous variable between 0 and 100

B. Binary variable taking 0 or 1

C. Unrestricted real variable

D. Percentage variable only

Correct Answer: B. Binary variable taking 0 or 1

Question 4

What does a feasible solution do?

A. Maximizes every possible objective

B. Satisfies every model constraint

C. Uses every resource completely

D. Always contains integer values

Correct Answer: B. Satisfies every model constraint

Question 5

In the Northstar case, what is the total cost of the stated shipment plan?

A. £4,900

B. £5,200

C. £5,600

D. £6,400

Correct Answer: C. £5,600

Question 6

Which optimization problem focuses on finding a least-cost route between an origin and a destination?

A. Shortest path

B. Maximum flow

C. Assignment

D. Minimum spanning tree

Correct Answer: A. Shortest path

Question 7

Why should objectives usually be normalized before using a weighted-sum model?

A. To force all variables to be integers

B. To make differently scaled objectives comparable

C. To remove all constraints

D. To guarantee a unique optimum

Correct Answer: B. To make differently scaled objectives comparable

Question 8

A constraint of the form y_A ≤ y_B commonly represents which logic?

A. Project A requires project B

B. Exactly one project must be selected

C. Project A and B cannot both be selected

D. Project B requires project A to be rejected

Correct Answer: A. Project A requires project B

Question 9

What is a key limitation of a shadow price?

A. It applies independently of all model assumptions

B. It is valid only within an appropriate sensitivity range

C. It always equals the market purchase price

D. It is defined only for binary variables

Correct Answer: B. It is valid only within an appropriate sensitivity range

Question 10

What is a Pareto-efficient solution?

A. A solution that maximizes every objective simultaneously

B. A solution that uses no constraints

C. A solution where improving one objective worsens at least one other objective

D. A solution that is always the cheapest

Correct Answer: C. A solution where improving one objective worsens at least one other objective