Operations Research and Decision Models

Introduction

Industrial engineering is concerned with designing and improving systems that use people, materials, machines, information, energy, and capital. Many operational problems involve competing objectives and scarce resources: a factory must decide its product mix, a transport manager must allocate shipments, a maintenance department must schedule technicians, and a hospital must balance service capacity against waiting time. Operations research (OR) provides a disciplined approach to such problems by representing a real system with a mathematical or computational model and identifying a good, feasible decision.

An OR study normally follows this sequence:

1. Define the problem, decision authority, objectives, and system boundaries.
2. Identify controllable decision variables and restrictions.
3. Construct a model using reliable data and justified assumptions.
4. Solve or simulate the model.
5. Validate the model and interpret the solution operationally.
6. Conduct sensitivity and risk analysis.
7. Implement, monitor, and revise the decision.

A mathematically optimal solution is not automatically a good managerial decision. It must be feasible, understandable, robust, ethical, and consistent with organizational constraints.

Learning Objectives

After studying this chapter, you should be able to:

• formulate linear, integer, transportation, assignment, and network models;
• solve small models and interpret objective values, constraints, and slack;
• distinguish certainty, risk, and uncertainty in decision analysis;
• use payoff tables, expected value, regret, and decision trees;
• explain sensitivity analysis and the meaning of shadow prices;
• describe queuing systems and calculate basic performance measures;
• explain the role of simulation when analytical models are inadequate;
• compare alternatives using multi-criteria decision-making;
• recognize modelling assumptions, implementation risks, and ethical responsibilities.

1. Model Formulation: From an Operational Problem to Mathematics

A model is a purposeful representation of a system. In a prescriptive OR model, the decision-maker selects values of variables to optimize an objective while satisfying constraints. The general optimization form is:

subject to:

and domain restrictions on the decision variables. Here, x_j is a decision variable, Z is the objective value, g_i represents a constraint, and b_i is the available resource or required level. Units must be consistent: if c_j is measured in dollars per unit and x_j in units, then c_jx_j is dollars.

A useful formulation distinguishes:

• Decision variables: what the organization controls.
• Parameters: known or estimated data, such as processing time or cost.
• Objective: the measure to maximize or minimize.
• Constraints: technological, resource, policy, safety, or demand limits.
• Feasible region: all decisions satisfying every constraint.
• Optimal solution: the feasible solution with the best objective value.

The assumptions should be recorded explicitly. For example, a linear production model assumes proportionality, additivity, divisibility where variables are continuous, and known coefficients. These assumptions may be acceptable for planning but unsuitable for detailed shop-floor scheduling.

2. Linear Programming

A linear programming (LP) model has a linear objective and linear constraints. A standard product-mix model is:

subject to:

where c_j is contribution per unit of product j, a_ij is the amount of resource i used by one unit of product j, and b_i is the available quantity of resource i. The inequalities may also be equalities or lower bounds.

Graphical interpretation

For two variables, each constraint can be plotted as a line. The feasible region is the intersection of all permissible half-planes. Under the assumptions of LP, if an optimum exists, at least one optimum occurs at an extreme point (corner point) of the feasible region. This is the basis of the graphical method and, in higher dimensions, the simplex method.

Worked example: product mix

A machine shop makes products A and B. Each A contributes $40 and uses 2 machine hours and 1 assembly hour. Each B contributes $30 and uses 1 machine hour and 2 assembly hours. Available capacity is 100 machine hours and 80 assembly hours. Demand limits are 40 units of A and 50 units of B.

Let x_A and x_B be production quantities. The model is:

subject to:

The intersection of the two capacity boundaries satisfies:

Solving gives x_A = 40 and x_B = 20. The objective value is $2,200. Relevant corner points include (0,0), (40,0), (40,20), and (0,40); their contributions are $0, $1,600, $2,200, and $1,200, respectively. Therefore, the optimum is 40 units of A and 20 units of B. Machine capacity is fully used, while assembly usage is 80 hours, also fully used.

Simplex and software interpretation

The simplex method moves between adjacent extreme points while improving the objective until no improving move exists. Modern solvers commonly use revised simplex, interior-point, or branch-and-bound methods. Software output must not be accepted without checking feasibility, units, objective coefficients, and whether the model represents the real system.

Dual values and sensitivity

The dual of a resource-constrained maximization model assigns a marginal value to each constraint. A shadow price estimates the improvement in objective value from one additional unit of a resource, within an allowable range and assuming other data remain fixed. It is not necessarily a market price and should not be extrapolated beyond the valid sensitivity interval.

Reduced cost indicates how much an objective coefficient of a currently zero variable would need to improve before that variable becomes attractive, subject to the solver’s sign convention. Sensitivity analysis may vary objective coefficients, resource capacities, demand limits, and technical coefficients. It identifies robust decisions and data requiring better measurement.

3. Integer and Mixed-Integer Models

Some decisions cannot be fractional. A crane, warehouse, project, or machine is indivisible; a setup decision is yes-or-no. Integer programming imposes:

Binary variables satisfy x_j ∈ {0,1}. A mixed-integer linear program (MILP) contains both continuous and integer variables.

Typical logical relationships include:

• If facility A is opened, facility B may be opened: y_B ≤ y_A.
• Product quantity is possible only if a line is activated: x ≤ My, where M is a valid upper bound.
• Exactly one option must be selected: y₁ + y₂ + … + y_k = 1.

The constant M should be as small as valid. An excessively large “big M” can cause numerical instability and weak computational performance.

Worked example: project selection

A company can select projects 1, 2, and 3. Their benefits are 80, 65, and 50 points; costs are $60, $45, and $30 thousand. The budget is $90 thousand, and at least one of projects 1 and 2 must be selected. Let y_i be 1 if project i is selected and 0 otherwise.

subject to:

Feasible selections include projects 1 and 2, with benefit 145 and cost $105 thousand, which is infeasible; projects 1 and 3, benefit 130 and cost $90 thousand; projects 2 and 3, benefit 115 and cost $75 thousand; and project 2 alone, benefit 65. Thus projects 1 and 3 are optimal among these combinations.

Integer models are solved using branch-and-bound, cutting planes, or branch-and-cut. Their computational difficulty can grow rapidly, so formulation quality and realistic bounds matter.

4. Transportation, Assignment, and Network Models

Transportation model

A transportation model allocates shipments from sources to destinations at minimum cost. Let x_ij be the quantity shipped from source i to destination j, c_ij the unit shipping cost, s_i source supply, and d_j destination demand.

subject to:

For a balanced model, total supply equals total demand and the inequalities are usually equalities. If supply exceeds demand, a dummy destination may represent unused supply; if demand exceeds supply, a dummy source may represent shortage, with an appropriate penalty cost.

Initial feasible solutions can be generated by the northwest-corner, least-cost, or Vogel approximation method. Optimality can then be tested using the stepping-stone or MODI method, or directly using an LP solver.

Assignment model

Assignment is a special case in which each agent is assigned to exactly one task and each task receives exactly one agent. With cost c_ij:

subject to:

The Hungarian algorithm solves the square assignment problem efficiently. Rectangular problems can be balanced with dummy rows or columns. Costs may represent time, risk, distance, or penalty rather than money.

Network models

A network consists of nodes and arcs. Important models include:

• Shortest path: minimize distance or cost from an origin to a destination.
• Minimum spanning tree: connect all nodes at minimum total link cost without cycles.
• Maximum flow: maximize movement through capacity-limited arcs.
• Minimum-cost flow: meet supplies and demands at minimum transportation cost.
• Project network: represent precedence relationships for scheduling.

For a shortest-path problem with arc costs c_ij, a path cost is the sum of its arc costs. Dijkstra’s algorithm applies when all arc costs are nonnegative. It repeatedly labels the unvisited node with the smallest temporary distance and relaxes adjacent arcs. Negative arc costs require other methods, such as Bellman–Ford, and negative cycles require careful interpretation.

5. Decision Models Under Certainty, Risk, and Uncertainty

A decision is made under certainty when the state of nature and consequences are known. Under risk, possible states and their probabilities are known or credibly estimated. Under uncertainty, probabilities are unavailable, disputed, or too unreliable to justify numerical use.

A payoff table lists alternatives, states of nature, and outcomes. If alternative a has payoff V_as in state s, then expected monetary value is:

where p_s is the probability of state s, with p_s ≥ 0 and ∑p_s = 1. EMV is appropriate when probabilities are meaningful and the decision can be repeated or the decision-maker is risk-neutral. It does not guarantee the best realized outcome.

Expected opportunity loss uses regret:

for maximization, where V*_s is the best payoff in state s. The expected regret is ∑p_sR_as. The expected value of perfect information is the expected payoff with perfect information minus the best current EMV. It is an upper bound on what information should cost.

When probabilities are not credible, criteria include:

• Maximax: choose the alternative with the largest possible payoff; optimistic.
• Maximin: choose the alternative with the best worst-case payoff; conservative.
• Minimax regret: minimize the largest possible regret.
• Laplace: assign equal probabilities when no state is judged more likely.
• Hurwicz: combine best and worst outcomes using an optimism coefficient.

These are decision rules, not universal truths. The appropriate rule depends on risk attitude, reversibility, strategic consequences, and stakeholder impact.

Decision trees

A decision tree displays decisions as squares, chance events as circles, and outcomes at terminal branches. Solve it by rollback: calculate expected values at chance nodes and select the preferred branch at decision nodes. Trees are especially useful for staged decisions, such as pilot testing followed by capacity expansion.

Figure: A decision tree separates controllable decisions from uncertain states and terminal payoffs.

6. Queuing Models

A queue forms when demand for service temporarily exceeds available service capacity. A queuing system consists of an arrival process, waiting line, service mechanism, queue discipline, and exit process. Common disciplines include first-come, first-served; priority; and shortest processing time.

The basic M/M/1 model assumes Poisson arrivals, exponentially distributed service times, one server, infinite waiting space, and first-come, first-served service. Let λ be the mean arrival rate and μ the mean service rate, both in customers per unit time. Stability requires:

where ρ is utilization. For the M/M/1 model:

Here L is the average number in the system, L_q the average number waiting, W the average time in the system, and W_q the average waiting time. Little’s law connects long-run averages:

The units verify: customers = customers/time × time. Little’s law applies broadly under stable long-run conditions, not only to M/M/1 systems.

For example, if λ = 6 customers/hour and μ = 8 customers/hour, then ρ = 0.75, L = 3 customers, L_q = 2.25 customers, W = 0.5 hour, and W_q = 0.375 hour. The results are averages, not guarantees for individual customers.

Analytical formulas become difficult when arrivals are scheduled, service times are highly variable, servers have different speeds, customers abandon the queue, or priorities interact. In such cases, simulation is often preferable.

7. Simulation and Experimentation

Simulation imitates the operation of a system over time. A discrete-event simulation changes state at events such as arrival, service completion, machine failure, or replenishment. A typical study is:

1. define objective and performance measures;
2. specify entities, resources, events, and state variables;
3. collect and fit input distributions;
4. verify logic and validate outputs;
5. choose warm-up, run length, and replications;
6. compare alternatives using confidence intervals and practical significance.

A random sample from a distribution should represent observed variability, not hide poor data. Replications with different random-number streams help estimate uncertainty. If an output has sample mean ȳ and sample standard deviation s from n independent replications, an approximate 95% confidence interval is:

where t₀.975,n−1 is the Student t critical value. This interval assumes independent replications and an approximately valid mean-based inference. A simulation is not a forecast merely because it produces precise decimal output; validity depends on model structure and input data.

8. Multi-Criteria Decision-Making

Operational decisions often involve cost, quality, safety, delivery, flexibility, energy, and employee effects. A single objective can conceal important consequences. Multi-criteria decision-making (MCDM) makes trade-offs explicit.

A weighted-sum model for alternative a is:

where w_k is the weight of criterion k, r_ak is the normalized rating of alternative a, and ∑w_k = 1. Benefit criteria should have higher ratings for better performance; cost criteria must be transformed so that lower cost receives a higher rating. Weights represent preferences, not physical constants.

The method is transparent but sensitive to normalization, weights, and criteria dependence. Best practice includes stakeholder participation, explicit definitions, sensitivity analysis, and reporting results before and after plausible weight changes. Methods such as the analytic hierarchy process, TOPSIS, outranking, and goal programming offer alternatives, but no method eliminates value judgments.

9. Illustrative Case: Capacity and Distribution at Apex Components

Background

Apex Components manufactures two precision brackets at two plants and supplies three regional warehouses. Demand varies seasonally, transport costs differ, and one finishing machine is approaching capacity. Management is considering overtime, subcontracting, and a second shift.

Problem

The immediate decision is to determine production quantities and shipments for the next month while minimizing production and transportation cost. A secondary decision is whether to activate overtime. The organization must also consider delivery reliability and worker fatigue.

Analysis

The engineering team first defines variables for production by plant and product, shipment by plant and warehouse, and a binary overtime variable. The model includes material availability, machine hours, labor hours, plant capacity, warehouse demand, and a logical constraint that overtime can be used only if authorized. A transportation submodel links plant shipments to warehouse demand. Because overtime is binary and production is continuous, the result is a MILP.

Before solving, analysts check that demand forecasts use the same time horizon as capacities, costs are in the same currency and period, and subcontracting capacity is real rather than nominal. They also create scenarios for high demand and machine downtime.

Approach and solution

The base model recommends normal production with no overtime, but the high-demand scenario activates overtime at Plant 2. Sensitivity analysis shows that the recommendation changes if demand for Warehouse 3 rises by more than 8%. A separate simulation of machine failures shows that the apparently cheaper plan creates unacceptable late deliveries when downtime is frequent. A robust policy therefore reserves limited overtime and pre-approves a subcontractor rather than committing all capacity to the nominal optimum.

Outcome

The final decision is not the lowest-cost base-case solution. It is a contingent policy: use normal production under ordinary demand, activate overtime when a monitored demand threshold is reached, and divert specified shipments if Plant 1 fails. This raises expected cost slightly but reduces service risk and avoids excessive emergency decisions.

Learning points

• A model boundary determines which costs and risks are visible.
• Mixed-integer decisions are appropriate for authorization and activation choices.
• Sensitivity and scenario analysis reveal whether an optimum is fragile.
• Simulation can expose congestion and downtime effects omitted from a static LP.
• Implementation requires ownership, triggers, data systems, and review rules.

10. Limitations, Governance, and Contemporary Practice

OR models simplify reality. Linear coefficients may ignore scale effects; forecasts may be biased; queues may not be stationary; and historical data may encode unequal treatment. An apparently neutral optimization model can allocate undesirable shifts disproportionately to a vulnerable workforce or prioritize profitable customers during shortages.

Professional practice requires:

• documenting assumptions, data sources, transformations, and model versions;
• testing infeasibility and extreme cases;
• separating validation data from calibration data where appropriate;
• protecting confidential employee, customer, and supplier information;
• explaining automated recommendations to affected decision-makers;
• preserving human accountability for safety-critical decisions;
• monitoring model drift and performance after implementation.

AI and machine learning can improve demand forecasting, anomaly detection, parameter estimation, and simulation metamodels. They do not replace optimization logic, domain knowledge, or governance. A forecast with an uncertainty interval is more useful than an unexplained point prediction. Hybrid systems increasingly combine machine-learning forecasts with constrained optimization, digital twins, real-time data, and reinforcement learning. Engineers must still verify feasibility, constraint compliance, cybersecurity, and ethical impact.

Best-Practice Checklist

1. Define the decision, not merely the symptom.
2. Use variables that correspond to actions someone can implement.
3. Keep units and time periods consistent.
4. Include all binding operational constraints and distinguish hard from soft constraints.
5. Validate data and test sensitivity to uncertain inputs.
6. Compare the model output with current practice and simple benchmarks.
7. Examine alternative scenarios, not only the base case.
8. Communicate trade-offs, shadow prices, risk, and limitations.
9. Pilot significant changes and monitor actual outcomes.
10. Retire or revise the model when the operating environment changes.

Chapter Summary

Operations research converts operational decisions into structured models. Linear programming optimizes a linear objective subject to linear constraints; integer and mixed-integer models represent indivisible and logical choices. Transportation and assignment models allocate resources efficiently, while network models address paths, flows, connectivity, and project relationships. Decision analysis uses payoff tables, expected value, regret, decision trees, and uncertainty criteria to compare alternatives. Sensitivity analysis identifies the effects of changing data and reveals whether a recommendation is robust. Queuing theory relates arrival rates, service rates, utilization, waiting, and congestion, while simulation models systems too complex for closed-form analysis. Multi-criteria methods make trade-offs among cost, service, quality, safety, and sustainability explicit. Effective OR practice combines mathematical quality with data governance, stakeholder involvement, ethical judgment, and implementation discipline.

Key Terms

Operations research: Systematic use of analytical models to support operational decisions.

Decision variable: A controllable quantity selected by the model.

Parameter: A fixed or estimated coefficient used in a model.

Feasible solution: A decision satisfying every model constraint.

Linear programming: Optimization with linear objective and constraint functions.

Shadow price: Marginal value of relaxing a constraint within a valid range.

Integer programming: Optimization requiring some or all variables to be integers.

Binary variable: Variable restricted to 0 or 1.

Transportation model: Network allocation model for shipping from sources to destinations.

Assignment model: One-to-one allocation of agents to tasks.

Network model: Model representing nodes and connecting arcs.

Payoff table: Matrix of alternatives, states, and outcomes.

Expected monetary value: Probability-weighted average payoff.

Regret: Loss relative to the best alternative after a state occurs.

Decision tree: Branching representation of sequential decisions and chance events.

Queue: Waiting line formed when demand for service is not immediately met.

Utilization: Fraction of service capacity being used.

Simulation: Computational imitation of system behavior over time.

Multi-criteria decision-making: Evaluation of alternatives across several criteria.

Sensitivity analysis: Study of how model results change when inputs vary.

Review Questions

Short Answer Questions

1. What are the five principal components of a linear programming formulation?
2. Why are binary variables useful in industrial engineering models?
3. Distinguish decision-making under risk from decision-making under uncertainty.
4. State Little’s law and explain its variables.
5. Why is sensitivity analysis important after obtaining an optimal solution?

Long Answer Questions

1. Explain the formulation and solution logic of a product-mix linear program, including feasible regions, extreme points, and shadow prices.
2. Compare transportation, assignment, and general network models, stating the operational situations for which each is appropriate.
3. Describe decision-tree rollback and explain when expected monetary value is an appropriate criterion.
4. Explain the assumptions of the M/M/1 queuing model and discuss situations in which those assumptions are unrealistic.
5. Discuss how simulation, optimization, and multi-criteria decision-making can be combined in a capacity-planning study.

Analytical / Application Questions

1. A workshop earns $55 per unit of Product X and $42 per unit of Product Y. X uses 3 machining hours and 1 finishing hour; Y uses 2 machining hours and 2 finishing hours. Available capacity is 120 machining hours and 80 finishing hours. Formulate the LP and determine the best production mix by evaluating relevant corner points.
2. A service desk receives 10 customers per hour and serves 12 customers per hour using one server. Calculate utilization, average number in the system, and average time in the system under the M/M/1 assumptions. Interpret the result operationally.
3. A manager must choose among three capacity plans under high, medium, and low demand. Construct a payoff table using estimated revenues and costs, then compare EMV, minimax regret, and maximin recommendations. Explain why the recommendations may differ.

Multiple Choice Questions with Answers

Question 1

Which condition is required for a stable M/M/1 queue?

A. λ = 0

B. λ > μ

C. λ < μ

D. λ = μ

Correct Answer: C. λ < μ

Question 2

In a linear program, a feasible solution is one that:

A. Has the largest objective value regardless of constraints

B. Satisfies all model constraints and variable restrictions

C. Uses every resource completely

D. Contains only integer variables

Correct Answer: B. Satisfies all model constraints and variable restrictions

Question 3

A binary variable is restricted to:

A. Any positive real value

B. −1 and 1

C. 0 and 1

D. All integers greater than 1

Correct Answer: C. 0 and 1

Question 4

For a maximization payoff table, regret is calculated as:

A. The selected payoff minus the best payoff in the state

B. The best payoff in the state minus the selected payoff

C. The probability multiplied by the payoff

D. The sum of all payoffs

Correct Answer: B. The best payoff in the state minus the selected payoff

Question 5

If λ = 6 customers/hour and μ = 8 customers/hour in an M/M/1 system, utilization is:

A. 0.25

B. 0.50

C. 0.75

D. 1.33

Correct Answer: C. 0.75

Question 6

Which model assigns each agent to exactly one task and each task to exactly one agent?

A. Assignment model

B. Unbalanced inventory model

C. Queuing model

D. Regression model

Correct Answer: A. Assignment model

Question 7

The objective of a transportation model is most commonly to:

A. Maximize queue length

B. Minimize total shipment cost

C. Maximize the number of decision variables

D. Eliminate all demand

Correct Answer: B. Minimize total shipment cost

Question 8

Expected monetary value requires:

A. Only the worst-case payoff

B. Equal payoffs in every state

C. Credible probabilities for possible states

D. All decision variables to be binary

Correct Answer: C. Credible probabilities for possible states

Question 9

Little’s law is expressed as:

A. L = λW

B. L = μ/λ

C. W = μλ

D. λ = L + W

Correct Answer: A. L = λW

Question 10

In a weighted-sum multi-criteria model, criterion weights should normally:

A. Sum to zero

B. Sum to one

C. All be greater than one

D. Be unrelated to stakeholder preferences

Correct Answer: B. Sum to one