Network Optimization and Scenario Analysis

Introduction

Distribution-network configuration determines where facilities are located, which customers they serve, how products move, and what capacity is reserved across the network. These decisions are strategic because facilities and contracts are often long-lived, while demand, transport prices, service expectations, regulations, and supply risks change continuously. A network that is efficient under one demand pattern may be expensive or fragile under another.

Optimization provides a disciplined method for comparing feasible configurations. It converts managerial choices into decision variables, expresses performance through objective functions, and represents physical, commercial, regulatory, and service requirements as constraints. Scenario analysis then tests whether the preferred design remains suitable when assumptions change. The result is not merely a mathematically optimal answer, but evidence for a defensible managerial decision.

This chapter develops linear programming (LP), mixed-integer linear programming (MILP), multi-objective optimization, sensitivity analysis, what-if analysis, stochastic programming, and robust optimization for distribution-network decisions.

Learning Objectives

After completing this chapter, the reader should be able to:

1. Formulate distribution-network decisions as LP and MILP models.
2. Define decision variables, objective functions, parameters, constraints, and units correctly.
3. Distinguish continuous, integer, binary, deterministic, stochastic, and robust formulations.
4. Apply weighted-sum, ε-constraint, and lexicographic methods to multiple objectives.
5. Interpret shadow prices, reduced costs, binding constraints, marginal values, and infeasibility.
6. Design scenario, sensitivity, what-if, stochastic, and robust analyses.
7. Translate model outputs into managerial recommendations while recognizing limitations and governance risks.

1. The Optimization Logic of Network Configuration

A distribution-network model has five elements:

• Sets and indices: facilities, customers, products, periods, transport modes, and scenarios.
• Parameters: demand, capacity, costs, distances, emissions factors, service times, and probabilities.
• Decision variables: facility opening, shipment quantities, inventory, mode selection, and customer allocation.
• Objective function: the quantity to minimize or maximize, such as total cost, profit, emissions, or service value.
• Constraints: requirements that feasible decisions must satisfy.

A general deterministic optimization model can be written as:

Here, x is the vector of decisions, f(x) is the objective, g_r(x) represents a constraint, b_r is its right-hand-side requirement, and X specifies domains such as non-negativity, integrality, or binary restrictions. An LP has linear objective and constraint expressions and continuous variables. An MILP retains linear expressions but allows some variables to be integer or binary.

Model quality depends more on formulation than on solver sophistication. A mathematically precise model with imperfect data can be improved through data collection; an incorrectly formulated model can produce confidently wrong decisions.

2. A Basic Facility-Location and Allocation Model

Consider potential distribution centres indexed by j ∈ J and customer zones indexed by i ∈ I. Let:

The total-cost MILP is:

subject to demand fulfillment:

facility capacity:

and domain restrictions:

The capacity constraint is a linking constraint. If y_j = 0, all shipments from facility j must be zero. If y_j = 1, shipments cannot exceed capacity. The model assumes divisible flows, known demand, one planning period, no inventory, and costs that are linear in volume. If customer demand can be split across facilities, equality is appropriate. If each customer must use exactly one facility, introduce binary assignment variables a_ij and impose:

with a_ij ∈ {0,1}. This increases model complexity but represents indivisible customer allocation.

Extensions

A realistic configuration model may include:

• minimum economic throughput: L_j y_j ≤ ∑_i x_ij;
• a maximum number of opened facilities: ∑_j y_j ≤ P;
• regional coverage: at least one facility within a service radius;
• product-specific capacity, handling, and compatibility;
• inbound procurement and outbound transport flows;
• inventory carried between periods;
• carbon limits or renewable-energy requirements;
• labour, dock, fleet, or storage constraints;
• business-continuity requirements, such as two qualified facilities for critical products.

For multiple periods t ∈ T, inventory balance is commonly represented as:

where I_jt is end-of-period inventory, q_kjt is inbound quantity from supplier or plant k, and x_ijt is outbound quantity. All terms must use the same product and time units. Initial inventory and final-inventory conditions must be stated explicitly.

3. Linear, Mixed-Integer, and Nonlinear Considerations

LP is appropriate when decisions are divisible and relationships are proportional. It is computationally efficient and provides useful dual information. MILP is required when the model includes yes/no decisions, minimum shipment sizes, vehicle counts, fixed charges, route activation, or assignment decisions.

A fixed-charge transport decision can be represented by a binary variable z_ij:

where U_ij is a valid upper bound. Choosing U_ij too large may weaken the formulation and slow computation; choosing it too small can remove feasible solutions. Tight, physically justified bounds are therefore important.

Some relationships are nonlinear. Examples include congestion, quantity discounts, economies of scale, and vehicle utilization. A piecewise-linear approximation can preserve MILP structure, but the approximation error should be measured. A model should not be linearized merely for convenience when the resulting decision materially distorts cost or service.

4. Objective Functions and Multi-Objective Optimization

A total-cost objective may include fixed facility cost, handling, inventory, transportation, shortage penalties, and emissions charges:

A carbon objective may be:

where e_ijm is emissions per unit shipped on lane i,j by mode m, and e_j^F is facility-related emissions associated with opening facility j. Units must be consistent, such as kilograms CO₂-equivalent per unit multiplied by units shipped.

Network decisions usually involve competing objectives: cost, response time, resilience, emissions, and employment. Three useful approaches are:

Weighted-sum method

Normalize objectives before combining them, because currency, hours, and tonnes are not directly comparable:

where w_k ≥ 0, ∑_k w_k = 1, and Ẑ_k is a normalized objective. A min–max normalization over a reference range can be written as:

The reference values must be documented; otherwise weights can create a false impression of precision.

ε-constraint method

Optimize one primary objective while imposing limits on others:

This is often easier to explain to managers because emissions and service become explicit policy thresholds.

Lexicographic optimization

Objectives are ranked. First minimize cost; among cost-optimal or near-optimal solutions, minimize emissions; then maximize resilience. This reflects strict priorities but may ignore small improvements in a higher-ranked objective unless tolerances are specified.

A solution is Pareto-efficient if no objective can improve without worsening at least one other objective. Presenting a Pareto frontier is generally more informative than reporting one supposedly universal optimum.

5. Scenario, Sensitivity, and What-If Analysis

These methods answer different questions.

• Scenario analysis: What happens under coherent alternative futures, such as high growth, fuel-price escalation, or a port disruption?
• Sensitivity analysis: How does the solution change when one parameter varies, often over a range?
• What-if analysis: What is the effect of a specific managerial action, such as opening a facility or imposing a service limit?
• Stress testing: Does the design remain viable under severe but plausible conditions?

A scenario should change a logically connected set of assumptions. For example, a rapid-growth scenario may alter demand, labour cost, transport congestion, and facility utilization together. Changing only one number may be useful for sensitivity analysis but is not necessarily a credible scenario.

A practical scenario workflow is:

1. Establish and validate the base case.
2. Identify uncertain or influential assumptions.
3. Construct internally consistent scenarios.
4. Solve the model for each scenario.
5. Compare cost, service, utilization, emissions, facility decisions, and unmet demand.
6. Examine whether the recommendation is stable.
7. Identify contingent actions and trigger thresholds.

Sensitivity interpretation in LP

For a linear program, the shadow price of a binding constraint estimates the marginal change in the optimal objective caused by a one-unit relaxation of its right-hand side, within the allowable sensitivity range. If a capacity constraint has a shadow price of 4 currency units per additional unit of capacity, one additional unit is worth approximately 4 currency units locally, provided the optimal basis does not change.

A nonbinding constraint commonly has zero shadow price, although this does not mean the constraint is unimportant under future scenarios. A reduced cost indicates the objective-coefficient improvement needed for a nonbasic variable to become attractive, subject to the solver’s sign convention and minimization/maximization formulation. Solver reports must therefore be read with their stated conventions.

MILP sensitivity is less straightforward. Binary changes can cause discontinuous shifts, and LP shadow prices from a relaxation do not automatically represent the value of a discrete decision. Use re-optimization, parametric runs, or controlled scenario experiments instead of treating MILP dual values as definitive managerial prices.

6. Stochastic Optimization

Deterministic models use one value for uncertain demand or cost. Stochastic programming represents a finite set of scenarios s ∈ S with probabilities p_s, where ∑_s p_s = 1. A two-stage model separates strategic first-stage decisions from scenario-dependent recourse decisions:

where y_j is selected before uncertainty is known and Q(y,s) is the optimized operating cost in scenario s. A deterministic-equivalent formulation may use shipment variables x_ijs for each scenario, linked to the common facility decisions y_j.

The expected-value objective is appropriate when probabilities are credible and risk neutrality is acceptable. It can produce a solution with attractive average cost but poor performance in a low-probability, high-impact scenario. Risk measures can be added. For loss [[MATH{L_s}]] in scenario s, expected shortfall at confidence level α can be represented using an auxiliary variable η and nonnegative variables u_s:

subject to:

Here, α is typically close to 1, such as 0.95. The formulation measures the average loss in the worst tail beyond the VaR threshold η, assuming a valid scenario probability model.

7. Robust Optimization

Robust optimization is useful when probability estimates are weak but uncertainty ranges are credible. Instead of optimizing expected performance, it seeks a solution that performs acceptably for all parameter values in a specified uncertainty set.

For uncertain demand d_i ∈ [d_i^L,d_i^U]], a conservative capacity requirement may impose:

This protects against simultaneous worst-case demand but can overbuild if all upper bounds are unlikely to occur together. Less conservative budgeted-uncertainty approaches permit only a limited number or magnitude of adverse deviations. The uncertainty set is therefore a managerial assumption, not a purely technical detail.

A robust design should be evaluated by its price of robustness: the additional cost paid for protection relative to the nominal optimum. Robustness can also be created operationally through postponement, flexible contracts, cross-training, alternate carriers, and scalable capacity rather than only through extra facilities.

8. Worked Example: Facility Selection Under Alternative Demand

Consider two candidate facilities, North and South, serving two customer zones, East and West. Annual demand is 600 and 400 units. Fixed costs are 24,000 and 20,000 currency units, and capacities are 700 and 600 units. Variable distribution costs are:

If only North opens, it cannot serve total demand of 1,000 units because capacity is 700. If only South opens, capacity is 600 and is also infeasible. Therefore both facilities must open under the stated demand and capacities. The least-cost allocation sends 600 East units from North and 400 West units from South. Total cost is:

The alternative allocation, with North serving West and South serving East, has variable cost:

compared with 19,200 for the selected allocation; hence it is inferior by 12,800 currency units in variable cost.

Now consider a high-growth scenario with East demand 800 and West demand 500. Total demand becomes 1,300 units, while combined capacity remains 1,300, so both facilities must operate at full capacity. The allocation must satisfy North’s 700-unit capacity and South’s 600-unit capacity. A natural allocation is North 700 units, including 600 East and 100 West, and South 400 East and 400 West. Variable cost is:

Adding fixed cost gives 79,700 currency units/year. The high-growth scenario therefore increases annual cost by 13,700 compared with the base case and eliminates spare capacity. Management should not interpret the base-case optimum as a complete recommendation; it should also consider whether expansion rights, overflow carriers, or temporary storage are available before capacity becomes binding.

9. Applied Case: Illustrative Omni-Channel Network Redesign

Background

The following is an illustrative case. Meridian Home Systems sells appliances through retailers and directly online. It operates one central warehouse and is considering two regional fulfilment centres. Online customers expect two-day delivery, while retailers accept four-day delivery. Demand is seasonal and returns are substantial.

Problem

The existing network has low fixed cost but high parcel distance and poor peak-season service. Management must decide whether to open regional facilities, how to allocate customer zones, and whether the environmental target should be a hard constraint or a weighted objective.

Analysis

The project team first cleansed order, freight, return, and labour data. It separated fixed facility costs from variable handling and transport costs, validated travel-time assumptions, and created three demand scenarios: base, 20% growth in online demand, and a peak-disruption scenario involving reduced carrier availability. The model included facility-opening binaries, customer allocation, product-family capacity, two-day service constraints for online orders, and a carbon ceiling.

Initial results showed that the cost-minimizing design opened only one regional facility, but this design failed the two-day requirement in the growth scenario. A weighted-sum model produced unstable results because small changes in the service weight caused a facility switch. The team therefore used an ε-constraint formulation: minimize total cost subject to at least 95% two-day online fulfilment and a specified emissions ceiling.

Approach and solution

The recommended design opened one regional facility immediately, retained the central warehouse, and secured an option on a second site. A flexible carrier contract provided peak capacity. The second site would be activated if online demand exceeded a trigger or if two-day fulfilment fell below the service threshold for two consecutive periods. The robust design cost more than the nominal cost-minimizing design but had substantially better performance under disruption.

Outcome

The managerial recommendation was not “open two facilities because the model says so.” It was a staged decision: invest in one facility, preserve expansion flexibility, monitor leading indicators, and pre-approve the second-stage action. This translated a static optimization result into a dynamic governance mechanism.

Learning points

1. Service constraints can be more informative than arbitrary service penalties.
2. Scenario-dependent recourse may justify flexible contracts rather than permanent capacity.
3. Model outputs must be converted into trigger rules and accountable decisions.
4. Data definitions, especially delivery-time and emissions boundaries, can materially change the result.

10. Interpreting and Governing Model Outputs

A solver’s optimum is conditional on the data, constraints, objective, and assumptions. Managers should inspect:

• total and component costs;
• facility openings and utilization;
• customer allocations and lane flows;
• service attainment and late orders;
• emissions and regulatory compliance;
• binding and near-binding constraints;
• unused capacity and flexibility;
• scenario stability;
• infeasibility or optimality gaps;
• assumptions with the greatest influence.

For MILP, an optimality gap may remain when the solver stops before proving exact optimality. If the best feasible solution is 10 million and the best bound is 9.8 million for a minimization problem, the relative gap is approximately:

The organization must decide whether the computational effort required to close that gap is worth the potential decision improvement.

Infeasibility should be investigated systematically. Common causes include insufficient total capacity, contradictory service requirements, incorrect units, overly tight minimum-throughput constraints, or accidental omission of an allowed source. Feasibility relaxation tools can identify which constraints require violation, but a relaxed solution is diagnostic, not automatically an acceptable operating plan.

Data, ethics, and governance

Optimization can embed unfair or harmful outcomes when service priorities systematically disadvantage remote, low-income, or protected communities. Constraints should reflect legal obligations and documented service policies rather than hidden discrimination. Emissions calculations must state system boundaries and emission factors; otherwise environmental claims may be misleading. Sensitive customer, labour, and supplier data require access controls, retention rules, and audit trails.

AI-assisted forecasting and optimization can improve scenario generation, but automated recommendations require human review. Forecast models should be monitored for drift, structural breaks, and biased training data. Decision owners should be able to explain why a facility was selected, which assumptions were used, and how exceptions are handled.

11. Best Practices and Limitations

Best practices include:

1. Define the decision horizon and planning granularity before modelling.
2. Use consistent units, currencies, time bases, and product definitions.
3. Separate parameters from decisions and document every source.
4. Validate the base model against known historical flows.
5. Use tight valid bounds and avoid unnecessary binary variables.
6. Test extreme cases to reveal formulation errors.
7. Report ranges and scenario results, not only one point estimate.
8. Include implementation constraints such as labour, permits, ramp-up, and contracts.
9. Reconcile model recommendations with qualitative risks and managerial knowledge.
10. Establish ownership for updating data and re-running scenarios.

Limitations remain. Linear costs may overlook congestion and service deterioration. Scenario probabilities may be subjective. Robust models can be excessively conservative. Static models may ignore path dependence, learning, and facility ramp-up. A mathematically optimal network may be impractical if change-management, labour relations, or permitting is ignored. Optimization should therefore support, not replace, structured managerial judgment.

Chapter Summary

Network optimization translates configuration choices into variables, objectives, and constraints. LP is suitable for divisible linear decisions; MILP represents facility openings, assignments, fixed charges, and other discrete choices. Facility-location models combine fixed costs, flow costs, demand satisfaction, and capacity restrictions. Multi-objective methods—including weighted sums, ε-constraints, and lexicographic optimization—make trade-offs between cost, service, resilience, and emissions explicit.

Scenario analysis evaluates coherent alternative futures, while sensitivity and what-if analysis examine parameter changes and managerial actions. LP dual values provide local marginal information, but MILP outputs require re-optimization because discrete changes are discontinuous. Stochastic programming uses scenario probabilities and recourse; robust optimization protects against uncertainty without relying entirely on probability estimates. Effective practice combines quantitative optimization with validation, governance, ethical review, operational flexibility, and trigger-based implementation.

Key Terms

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

Mixed-integer linear programming: Linear optimization containing integer or binary variables.

Decision variable: A controllable quantity selected by the model.

Parameter: A model input treated as known or estimated.

Objective function: Mathematical representation of the performance measure being minimized or maximized.

Constraint: A condition that feasible decisions must satisfy.

Linking constraint: A constraint connecting a continuous activity to a binary activation decision.

Shadow price: Local marginal value of relaxing a constraint in an LP.

Reduced cost: Objective-coefficient improvement required for a nonbasic variable to become attractive, subject to solver conventions.

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

ε-constraint method: Optimization of one objective while bounding other objectives.

Scenario analysis: Evaluation of the model under coherent alternative assumptions.

Sensitivity analysis: Examination of solution changes caused by parameter variation.

Stochastic programming: Optimization that explicitly represents uncertain outcomes and probabilities.

Recourse: Subsequent decisions made after uncertainty is observed.

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

CVaR: Conditional Value at Risk; average loss in the worst tail beyond a selected confidence threshold.

Optimality gap: Difference between a feasible solution and a mathematical bound on the optimum.

Price of robustness: Additional cost incurred to protect against uncertainty.

Review Questions

Short Answer Questions

1. What distinguishes an LP from an MILP?
2. Why is the constraint ∑_i x_ij ≤ K_j y_j important in facility-location modelling?
3. Define a Pareto-efficient solution.
4. How does scenario analysis differ from sensitivity analysis?
5. What does a shadow price represent in a linear program?

Long Answer Questions

1. Formulate and explain a facility-location MILP containing facility fixed costs, customer demand, capacity, and shipment costs.
2. Compare weighted-sum, ε-constraint, and lexicographic approaches to multi-objective network design.
3. Explain the difference between stochastic and robust optimization and identify circumstances in which each is appropriate.
4. Discuss why MILP sensitivity analysis cannot be interpreted in the same way as LP shadow-price analysis.
5. Explain how data quality, governance, ethics, and implementation constraints affect the credibility of optimization results.

Analytical / Application Questions

1. A company has three candidate facilities, two customer zones, fixed opening costs, and facility capacities. Design an MILP and explain how you would test the result under 10%, 20%, and 30% demand growth.
2. A cost-minimizing network meets only 88% of a two-day delivery target, while a higher-cost network meets 97%. Recommend an objective or constraint approach and justify the choice.
3. Demand is uncertain, probabilities are unreliable, and management can tolerate at most 5% unmet demand in any plausible case. Propose a robust or hybrid modelling approach and explain the uncertainty set and decision triggers.

Multiple Choice Questions with Answers

Question 1

Which variable type is required to represent whether a distribution centre is opened?

A. Continuous variable

B. Binary variable

C. Unrestricted real variable

D. Probability variable

Correct Answer: B. Binary variable

Question 2

In a facility-location model, what does the constraint ∑_i x_ij ≤ K_j y_j ensure?

A. Every customer is assigned to exactly two facilities

B. A closed facility cannot ship and an open facility cannot exceed capacity

C. All facilities operate at full capacity

D. Transportation cost is zero

Correct Answer: B. A closed facility cannot ship and an open facility cannot exceed capacity

Question 3

Which method minimizes one primary objective while imposing limits on other objectives?

A. ε-constraint method

B. Random sampling

C. Unrestricted relaxation

D. Moving average method

Correct Answer: A. ε-constraint method

Question 4

If two facilities have combined capacity exactly equal to total demand, what is the likely implication?

A. Both can remain closed

B. At least one facility must have zero utilization

C. Both facilities may be required and spare capacity is zero

D. Demand constraints become irrelevant

Correct Answer: C. Both facilities may be required and spare capacity is zero

Question 5

What does a shadow price generally estimate in an LP?

A. The fixed cost of opening a facility

B. The local marginal objective change from relaxing a constraint

C. The probability of demand failure

D. The number of binary variables

Correct Answer: B. The local marginal objective change from relaxing a constraint

Question 6

What distinguishes a stochastic optimization model from a nominal deterministic model?

A. It excludes all uncertainty

B. It represents alternative uncertain outcomes, often with probabilities

C. It requires every variable to be binary

D. It cannot include capacity constraints

Correct Answer: B. It represents alternative uncertain outcomes, often with probabilities

Question 7

Robust optimization is particularly useful when:

A. All future probabilities are known exactly

B. Uncertainty ranges are credible but probability estimates are weak

C. There are no constraints

D. Only historical costs matter

Correct Answer: B. Uncertainty ranges are credible but probability estimates are weak

Question 8

For a minimization problem with feasible solution 10.0 million and bound 9.8 million, the approximate relative optimality gap is:

A. 0.2%

B. 2%

C. 9.8%

D. 19.8%

Correct Answer: B. 2%

Question 9

Which statement about a nonbinding LP constraint is generally correct?

A. Its local shadow price is commonly zero

B. It must be violated

C. It determines every facility location

D. It always has the highest marginal value

Correct Answer: A. Its local shadow price is commonly zero

Question 10

Why should objectives with different units be normalized before a weighted sum?

A. To make all variables binary

B. To make incomparable scales more interpretable in the combined objective

C. To eliminate all uncertainty

D. To guarantee a global integer optimum

Correct Answer: B. To make incomparable scales more interpretable in the combined objective