8.1 Optimization Modeling: Decision Variables, Objectives, and Constraints

FE Industrial and Systems Study Guide Exam

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1 Chapter 1: Exam Overview, CBT Strategy & Handbook Navigation

1.1 FE Industrial Exam Structure, CBT Logistics, and Pacing 1.2 Using the NCEES FE Reference Handbook and Approved Calculators

2 Chapter 2: Mathematics for Industrial Engineers

2.1 Analytic Geometry, Coordinate Systems, and Trigonometry 2.2 Calculus: Differential, Integral, and Series Approximations 2.3 Linear Algebra: Matrix Operations, Systems of Equations, and Vectors

3 Chapter 3: Engineering Sciences Core

3.1 Thermodynamics Fundamentals and Fluid Flow Principles 3.2 Statics, Dynamics, and Mechanics of Materials Core 3.3 Electrical Circuits, DC/AC Analysis, and Power in Industrial Systems

4 Chapter 4: Ethics and Professional Practice

4.1 NCEES Model Rules of Professional Conduct and Technical Society Ethics 4.2 Engineering Licensure, Public Welfare, and Professional Liability 4.3 Engineering Contracts, Intellectual Property, and Conflicts of Interest

5 Chapter 5: Engineering Economics

5.1 Time Value of Money, Discount Factors, and Equivalence 5.2 Evaluating Alternatives: Present Worth, Equivalent Annual Cost, and IRR 5.3 Incremental Rate of Return, Replacement Analysis, and Break-Even 5.4 Depreciation Methods, MACRS, Income Taxes, and Inflation

6 Chapter 6: Probability Foundations & Distributions

6.1 Basic Probability Concepts, Conditional Probability, and Bayes' Theorem 6.2 Discrete Probability Distributions 6.3 Continuous Probability Distributions

7 Chapter 7: Statistical Inference, Regression & Design of Experiments

7.1 Sampling Distributions, Central Limit Theorem, and Confidence Intervals 7.2 Hypothesis Testing: Means, Proportions, and Variances 7.3 Linear Regression, Correlation Analysis, and Design of Experiments (ANOVA)

8 Chapter 8: Data, Logic, Modeling & Linear Programming

8.1 Optimization Modeling: Decision Variables, Objectives, and Constraints 8.2 Linear Programming: Graphical Solutions and Simplex Fundamentals 8.3 Duality, Sensitivity Analysis, and Integer Linear Programming 8.4 Data, Logic Development, and Analytics

9 Chapter 9: Stochastic Models & Simulation

9.1 Queuing Theory: Little's Law, M/M/1, and Multi-Server Systems 9.2 Discrete-Time Markov Chains: Transition Matrices and Steady-State Probabilities 9.3 Discrete-Event Simulation Principles and Monte Carlo Modeling

10 Chapter 10: Engineering Management, Project Planning & Performance Measurement

10.1 Organizational Structure, Leadership, and Worker Motivation Theories 10.2 Project Management: WBS, Critical Path Method (CPM), PERT, and Crashing 10.3 Earned Value Management (EVM), Decision Trees, and Risk Preference 10.4 Performance Measurement: Productivity, KPIs, Wage Systems, and the Balanced Scorecard

11 Chapter 11: Manufacturing & Production Systems

11.1 Manufacturing Processes, Material Processing, and Tolerancing 11.2 Assembly Line Balancing, Takt Time, Cycle Time, and Throughput 11.3 Sustainable Manufacturing, Waste Minimization, and Life-Cycle Assessment

12 Chapter 12: Production Planning, Inventory Control & Sequencing

12.1 Demand Forecasting Methods and Forecast Error Tracking 12.2 Aggregate Planning, Master Production Scheduling (MPS), and MRP 12.3 Deterministic Inventory Models: EOQ, EPQ, and Safety Stock 12.4 Job Sequencing, Dispatching Rules, and Flow Shop Scheduling

13 Chapter 13: Lean Systems & Theory of Constraints

13.1 Lean Foundations: Value Stream Mapping, Seven Wastes, and 5S 13.2 Pull Systems, Kanban, Just-In-Time (JIT), and SMED Setup Reduction 13.3 Theory of Constraints: Bottleneck Identification and Drum-Buffer-Rope

14 Chapter 14: Facilities Planning & Material Handling

14.1 Facility Location Analysis: Qualitative Factor Rating and Centroid Method 14.2 Facility Layout Design and Flow Analysis 14.3 Material Handling Equipment, Warehousing, and AS/RS

15 Chapter 15: Supply Chain Design & Logistics

15.1 Supply Chain Network Architecture and Transportation Modes 15.2 Distribution Strategies, Cross-Docking, and Vehicle Routing 15.3 The Bullwhip Effect, Risk Pooling, and Supply Chain Coordination

16 Chapter 16: Human Factors & Cognitive Ergonomics

16.1 Cognitive Engineering, Human-Machine Interface, Displays, and Controls 16.2 Anthropometric Principles, Percentile Design, and Biomechanical Posture 16.3 Environmental Ergonomics: Industrial Noise, Lighting, and Heat Stress

17 Chapter 17: Physical Ergonomics & Workplace Safety

17.1 Physical Ergonomics: The NIOSH Manual Lifting Equation 17.2 Cumulative Trauma Disorders and Postural Assessment 17.3 Workplace Safety: OSHA Standards, Hazard Analysis, and Control Hierarchy

18 Chapter 18: Work Design & Measurement

18.1 Methods Engineering, Process Charting, and Principles of Motion Economy 18.2 Direct Stopwatch Time Study: Observed Time, Performance Rating, and Allowances 18.3 Predetermined Time Systems, Work Sampling, and Learning Curves

19 Chapter 19: Quality Management, Six Sigma & Analytical Tools

19.1 Quality Philosophies, Total Quality Management, and Cost of Quality 19.2 Six Sigma DMAIC Methodology and Quality Function Deployment 19.3 The Seven Basic Quality Tools, Root Cause Analysis, and Taguchi Loss Function

20 Chapter 20: Statistical Process Control

20.1 Statistical Process Control Principles, Variation, and Run Rules 20.2 Variable Control Charts: X-bar & R, X-bar & S, and Individual-Moving Range 20.3 Attribute Control Charts: Defectives (p, np) and Defects (c, u)

21 Chapter 21: Process Capability & Acceptance Sampling

21.1 Process Capability Indices and Defect Levels 21.2 Acceptance Sampling Concepts and Operating Characteristic Curves 21.3 Sampling Risks, AQL, LTPD, and Standard Sampling Plans

22 Chapter 22: Systems Engineering & Life-Cycle Management

22.1 The Systems Engineering V-Model, Stakeholder Requirements, and Functional Architecture 22.2 Configuration Management, Interface Control, and Verification & Validation 22.3 Life-Cycle Costing, Maintainability, and System Availability

23 Chapter 23: Reliability Engineering & Risk Analysis

23.1 Reliability Mathematics, Hazard Rates, and the Bathtub Curve 23.2 System Reliability Configurations: Series, Parallel, and k-out-of-n Systems 23.3 Failure Modes and Effects Analysis and Fault Tree Analysis

8.1 Optimization Modeling: Decision Variables, Objectives, and Constraints

Key Takeaways Explain This Topic AI

Every mathematical optimization model requires three structural components: decision variables (controllable inputs), a single scalar objective function (criterion to maximize or minimize), and structural constraints with explicit non-negativity restrictions.

Linear programming relies on four indispensable assumptions: proportionality (linear scaling without quantity discounts), additivity (no cross-variable interaction terms), divisibility (continuous fractional values permitted), and certainty (deterministic parameters).

Fractional blending specifications (e.g., octane ratings, alloy purities) must be linearized by cross-multiplying the denominator across the inequality before solving.

Multi-period production scheduling models enforce inter-temporal physical continuity through inventory conservation balance equations: I_t = I_{t-1} + P_t - D_t.

In transportation and transshipment network models, every intermediate transfer point enforces strict flow conservation where total incoming flow equals total outgoing flow (Inflow - Outflow = 0).

Last updated: September 2026

8.1 Optimization Modeling: Decision Variables, Objectives, and Constraints

Mathematical programming is the cornerstone of operations research and industrial systems engineering. In modern production facilities, logistics networks, and supply chains, engineers routinely face complex decisions regarding how to allocate scarce resources—such as machine hours, labor shifts, raw materials, capital, and warehouse space—to achieve an optimal operational objective. The NCEES FE Industrial and Systems examination tests your ability to translate verbal problem descriptions into clean, rigorous mathematical formulations and evaluate their underlying assumptions.

1. The Three Pillars of Mathematical Programming

Every mathematical optimization model, regardless of scale or complexity, is constructed upon three indispensable elements: ┌────────────────────────────────────────┐

• │ MATHEMATICAL PROGRAM │
• └───────────────────┬────────────────────┘
• │
• ┌──────────────────────────────┼──────────────────────────────┐
• ▼ ▼ ▼
• ┌─────────────────┐ ┌───────────────────┐ ┌────────────────────┐
• │ 1. DECISION │ │ 2. OBJECTIVE │ │ 3. CONSTRAINTS │
• │ VARIABLES │ │ FUNCTION │ │ │
• │ What decisions │ │ Single criterion │ │ Resource limits, │
• │ must be made? │ │ to max or min │ │ demand, bounds │
• │ (x_1, x_2,...) │ │ Max/Min Z = c^T x│ │ Ax <= b, x >= 0 │
• └─────────────────┘ └───────────────────┘ └────────────────────┘
• 1. Decision Variables ( x j x_j x j ​ )

Decision variables represent the quantifiable choices under the direct control of the decision-maker. Formulating an optimization model always begins by defining decision variables with precise units and explicit temporal horizons. Example: x j = number of units of product j produced per week [units/week] \text{Example: } x_j = \text{number of units of product } j \text{ produced per week [units/week]} Example: x j ​ = number of units of product j produced per week [units/week]

Exam Watchout : Never define a decision variable vaguely (e.g., "let x 1 = Product 1 x_1 = \text{Product 1} x 1 ​ = Product 1 "). A decision variable must represent a numerical quantity with explicit units of measurement (e.g., " x 1 = gallons of chemical additive A blended per operating shift x_1 = \text{gallons of chemical additive A blended per operating shift} x 1 ​ = gallons of chemical additive A blended per operating shift ").

2. Objective Function ( Z Z Z )

The objective function expresses the single operational performance measure to be optimized as a function of the decision variables. In linear programming, this function is strictly a linear combination of the decision variables: Maximize or Minimize Z = ∑ j = 1 n c j x j = c 1 x 1 + c 2 x 2 + ⋯ + c n x n \text{Maximize or Minimize } Z = \sum_{j=1}^{n} c_j x_j = c_1 x_1 + c_2 x_2 + \dots + c_n x_n Maximize or Minimize Z = j = 1 ∑ n ​ c j ​ x j ​ = c 1 ​ x 1 ​ + c 2 ​ x 2 ​ + ⋯ + c n ​ x n ​

Where c j c_j c j ​ represents the objective function coefficient (e.g., unit profit contribution, unit revenue, unit acquisition cost, or unit cycle time) associated with activity j j j .

3. Constraints

Constraints represent technological limitations, physical capacities, legal regulations, or contractual requirements that restrict the values the decision variables can assume. They are categorized into:

Structural (Technological) Constraints : Functional restrictions imposed by limited resources: ∑ j = 1 n a i j x j ≤ b i (resource capacity limitations) \sum_{j=1}^{n} a_{ij} x_j \le b_i \quad \text{(resource capacity limitations)} j = 1 ∑ n ​ a ij ​ x j ​ ≤ b i ​ (resource capacity limitations) ∑ j = 1 n a i j x j ≥ b i (minimum service levels or contractual requirements) \sum_{j=1}^{n} a_{ij} x_j \ge b_i \quad \text{(minimum service levels or contractual requirements)} j = 1 ∑ n ​ a ij ​ x j ​ ≥ b i ​ (minimum service levels or contractual requirements) ∑ j = 1 n a i j x j = b i (exact physical blending or balance requirements) \sum_{j=1}^{n} a_{ij} x_j = b_i \quad \text{(exact physical blending or balance requirements)} j = 1 ∑ n ​ a ij ​ x j ​ = b i ​ (exact physical blending or balance requirements)

Where a i j a_{ij} a ij ​ is the technological consumption coefficient (units of resource i i i consumed per unit of activity j j j ), and b i b_i b i ​ is the right-hand-side (RHS) available capacity or requirement.

Sign Restrictions (Non-Negativity) : x j ≥ 0 ∀ j = 1 , 2 , … , n x_j \ge 0 \quad \forall j = 1, 2, \dots, n x j ​ ≥ 0 ∀ j = 1 , 2 , … , n

Negative production or negative inventory transfers are physically meaningless in standard industrial settings. If a variable is allowed to take positive, zero, or negative values (e.g., net cash flow or coordinate deviations), it must be explicitly defined as Unrestricted in Sign (URS) and represented as the difference between two non-negative variables: x j = x j + − x j − x_j = x_j^+ - x_j^- x j ​ = x j + ​ − x j − ​ , where x j + , x j − ≥ 0 x_j^+, x_j^- \ge 0 x j + ​ , x j − ​ ≥ 0 .

2. Fundamental Assumptions of Linear Programming

For a mathematical model to qualify as a valid Linear Program (LP) , four core mathematical assumptions must hold simultaneously:

Assumption Mathematical Meaning Practical Industrial Meaning Common Violation Trap

Proportionality The contribution of each variable to the objective and constraints is strictly proportional to its value: f ( k x j ) = k f ( x j ) f(k x_j) = k f(x_j) f ( k x j ​ ) = k f ( x j ​ ) . Constant returns to scale. No setup costs, no quantity discounts, no learning curve efficiencies. Bulk purchasing discounts where unit material cost drops after buying 1,000 kg.

Additivity The total objective value and total resource consumption are the exact sum of individual contributions: f ( x 1 + x 2 ) = f ( x 1 ) + f ( x 2 ) f(x_1 + x_2) = f(x_1) + f(x_2) f ( x 1 ​ + x 2 ​ ) = f ( x 1 ​ ) + f ( x 2 ​ ) . No cross-variable interaction effects, synergies, or chemical interference. Synergistic effects where combining two chemical additives yields greater yield than their sum ( x 1 x 2 x_1 x_2 x 1 ​ x 2 ​ ).

Divisibility Decision variables can assume any non-negative real value: x j ∈ R + x_j \in \mathbb{R}^+ x j ​ ∈ R + . Fractional solutions (e.g., 4.37 4.37 4.37 hours or 12.5 12.5 12.5 liters) are physically acceptable and implementable. Indivisible items like airplanes, cargo ships, or personnel schedules requiring integer counts.

Certainty Every parameter ( c j , a i j , b i c_j, a_{ij}, b_i c j ​ , a ij ​ , b i ​ ) is known with deterministic certainty. No stochastic variation, no demand volatility, and no machine breakdown probabilities. Fluctuating market prices or random machine failure rates modeled with mean values. ┌───────────────────────────────────┐

• │ LP CORE ASSUMPTIONS │
• └─────────────────┬─────────────────┘
• ┌─────────────────┬─────────┴─────────┬─────────────────┐
• ▼ ▼ ▼ ▼
• ┌───────────────┐ ┌───────────────┐ ┌───────────────┐ ┌───────────────┐
• │PROPORTIONALITY│ │ ADDITIVITY │ │ DIVISIBILITY │ │ CERTAINTY │
• │ c_j is fixed │ │ No cross-terms│ │ Continuous │ │ Deterministic │
• │ regardless │ │ (no x_1*x_2) │ │ fractional │ │ parameters │
• │ of scale │ │ Sum of parts │ │ values valid │ │ (no randoms) │
• └───────────────┘ └───────────────┘ └───────────────┘ └───────────────┘
• 3. Standard Formulation Taxonomy & Problem Archetypes

The FE Industrial exam frequently presents optimization scenarios drawn from four classic problem classes. Recognizing these patterns allows you to establish the correct decision variables and constraints in seconds.

1. Product Mix Models

A facility manufactures n n n distinct products using m m m shared, limited resources (machine capacities, skilled labor hours, raw material inventories). The goal is to determine the production volume of each product to maximize total contribution margin. Maximize Z = ∑ j = 1 n c j x j Subject to: ∑ j = 1 n a i j x j ≤ b i , ∀ i = 1 , … , m x j ≥ 0 , ∀ j = 1 , … , n \begin{aligned}

• \text{Maximize } & Z = \sum_{j=1}^{n} c_j x_j \\
• \text{Subject to: } & \sum_{j=1}^{n} a_{ij} x_j \le b_i, \quad \forall i = 1, \dots, m \\
• & x_j \ge 0, \quad \forall j = 1, \dots, n

\end{aligned} Maximize Subject to: ​ Z = j = 1 ∑ n ​ c j ​ x j ​ j = 1 ∑ n ​ a ij ​ x j ​ ≤ b i ​ , ∀ i = 1 , … , m x j ​ ≥ 0 , ∀ j = 1 , … , n ​

2. Blending and Quality Specification Models

Blending problems require determining the optimal proportions of raw feedstocks (e.g., crude oils, chemical reagents, metal alloys, animal feeds) to produce an end product that meets strict physical, chemical, or nutritional property thresholds at minimum total cost.

A common challenge on the FE exam is linearizing ratio and percentage constraints . Suppose feedstock j j j has a property rating q j q_j q j ​ (such as octane number or sulfur percentage) and costs c j c_j c j ​ per gallon. The blended fuel must have a minimum weighted property rating of Q min ⁡ Q_{\min} Q m i n ​ : ∑ j = 1 n q j x j ∑ j = 1 n x j ≥ Q min ⁡ \frac{\sum_{j=1}^{n} q_j x_j}{\sum_{j=1}^{n} x_j} \ge Q_{\min} ∑ j = 1 n ​ x j ​ ∑ j = 1 n ​ q j ​ x j ​ ​ ≥ Q m i n ​

As written, this constraint is non-linear because decision variables appear in both the numerator and denominator. Because ∑ x j > 0 \sum x_j > 0 ∑ x j ​ > 0 , we cross-multiply the denominator and collect like terms to produce a strictly linear constraint: ∑ j = 1 n q j x j ≥ Q min ⁡ ∑ j = 1 n x j ⟹ ∑ j = 1 n ( q j − Q min ⁡ ) x j ≥ 0 \sum_{j=1}^{n} q_j x_j \ge Q_{\min} \sum_{j=1}^{n} x_j \implies \sum_{j=1}^{n} (q_j - Q_{\min}) x_j \ge 0 j = 1 ∑ n ​ q j ​ x j ​ ≥ Q m i n ​ j = 1 ∑ n ​ x j ​ ⟹ j = 1 ∑ n ​ ( q j ​ − Q m i n ​ ) x j ​ ≥ 0

3. Transportation and Transshipment Networks

In a classical Transportation Problem , a commodity must be shipped from m m m supply origins (factories, warehouses) with capacities s i s_i s i ​ to n n n demand destinations (retailers, regional depots) with requirements d j d_j d j ​ at minimum total transit cost. Decision variables are defined as: x i j = units shipped from origin i to destination j x_{ij} = \text{units shipped from origin } i \text{ to destination } j x ij ​ = units shipped from origin i to destination j Minimize Z = ∑ i = 1 m ∑ j = 1 n c i j x i j Subject to: ∑ j = 1 n x i j ≤ s i , ∀ i = 1 , … , m (Supply constraints) ∑ i = 1 m x i j ≥ d j , ∀ j = 1 , … , n (Demand constraints) x i j ≥ 0 , ∀ i , j \begin{aligned}

\text{Minimize } & Z = \sum_{i=1}^{m} \sum_{j=1}^{n} c_{ij} x_{ij} \\

\text{Subject to: } & \sum_{j=1}^{n} x_{ij} \le s_i, \quad \forall i = 1, \dots, m \quad \text{(Supply constraints)} \\

• & \sum_{i=1}^{m} x_{ij} \ge d_j, \quad \forall j = 1, \dots, n \quad \text{(Demand constraints)} \\
• & x_{ij} \ge 0, \quad \forall i, j

\end{aligned} Minimize Subject to: ​ Z = i = 1 ∑ m ​ j = 1 ∑ n ​ c ij ​ x ij ​ j = 1 ∑ n ​ x ij ​ ≤ s i ​ , ∀ i = 1 , … , m (Supply constraints) i = 1 ∑ m ​ x ij ​ ≥ d j ​ , ∀ j = 1 , … , n (Demand constraints) x ij ​ ≥ 0 , ∀ i , j ​

In a Transshipment Problem , intermediate transshipment nodes (cross-docks, consolidation centers) exist where goods enter and leave. For every intermediate node k k k , a flow conservation constraint must hold: ∑ all i x i k − ∑ all j x k j = 0 ⟺ Total Inflow − Total Outflow = 0 \sum_{\text{all } i} x_{ik} - \sum_{\text{all } j} x_{kj} = 0 \quad \iff \quad \text{Total Inflow} - \text{Total Outflow} = 0 all i ∑ ​ x ik ​ − all j ∑ ​ x kj ​ = 0 ⟺ Total Inflow − Total Outflow = 0

4. Production Planning and Multi-Period Scheduling Models

Manufacturing systems must plan production over a discrete planning horizon of T T T time periods (weeks or months) with fluctuating seasonal demand D t D_t D t ​ . In each period t t t , the system can produce units using regular time ( P t P_t P t ​ ), overtime ( O t O_t O t ​ ), or subcontracting ( S t S_t S t ​ ), and carry unsold units into future periods as inventory ( I t I_t I t ​ ).

The fundamental governing relationship is the Inventory Balance Equation , which enforces physical continuity across successive time periods: I t = I t − 1 + P t + O t + S t − D t , ∀ t = 1 , … , T I_{t} = I_{t-1} + P_t + O_t + S_t - D_t, \quad \forall t = 1, \dots, T I t ​ = I t − 1 ​ + P t ​ + O t ​ + S t ​ − D t ​ , ∀ t = 1 , … , T

Rearranging to standard linear form with decision variables on the left and known parameters on the right: I t − 1 + P t + O t + S t − I t = D t I_{t-1} + P_t + O_t + S_t - I_t = D_t I t − 1 ​ + P t ​ + O t ​ + S t ​ − I t ​ = D t ​

Where I 0 I_0 I 0 ​ is the known initial inventory at the start of the planning horizon.

4. Step-by-Step Methodology for Converting Word Problems into LP Formulations

When confronting a complex verbal optimization question on the FE exam, follow this systematic five-step protocol:

Step 1: Identify the Decisions and Define Variables Ask: "What numerical decisions must be made to solve this problem?" Write down each variable x j x_j x j ​ with explicit units (e.g., units/hour, kg/batch, dollars invested).

Step 2: Formulate the Objective Function Ask: "What is the overarching operational goal?" Determine whether the goal is maximization (profit, revenue, efficiency, yield) or minimization (cost, scrap, cycle time, transit distance). Express Z Z Z as a linear combination of the decision variables.

Step 3: Tabulate Resource Limits and Draft Structural Constraints Identify every limiting factor mentioned in the prompt. Group them into logical categories: capacity limits ( ≤ \le ≤ ), minimum production/demand targets ( ≥ \ge ≥ ), and exact balance/conservation equations ( = = = ). Ensure that each constraint has variable terms on the LHS and pure constants on the RHS.

Step 4: State Non-Negativity and Variable Bounds Explicitly specify x j ≥ 0 x_j \ge 0 x j ​ ≥ 0 for all variables, or indicate any unrestricted/bounded variables ( L j ≤ x j ≤ U j L_j \le x_j \le U_j L j ​ ≤ x j ​ ≤ U j ​ ).

Step 5: Perform Dimensional Consistency Analysis Verify that the units of measurement on both sides of every constraint match exactly. If the RHS represents hours, the LHS must sum to hours (e.g., hours/unit × units = hours \text{hours/unit} \times \text{units} = \text{hours} hours/unit × units = hours ). If cycle times are given in minutes, convert either the LHS or the RHS so that time bases are uniform.

• 5. Step-by-Step Worked Engineering Examples
• Worked Example 8.1.1: Product Mix in an Advanced Electronics Facility

Problem : An industrial electronics plant manufactures two types of programmable logic controllers: Standard ( x 1 x_1 x 1 ​ ) and Industrial ( x 2 x_2 x 2 ​ ). Each controller must pass through three sequential processing centers: Surface Mount Technology (SMT) Placement, Automated Optical Inspection (AOI), and Environmental Stress Testing. The processing times per unit, total available weekly machine capacities, and net unit profit contributions are summarized below:

Work Center Standard ( x 1 x_1 x 1 ​ ) [hours/unit] Industrial ( x 2 x_2 x 2 ​ ) [hours/unit] Total Available Capacity [hours/week]

• SMT Placement 1.5 2.5 120
• AOI Inspection 0.8 1.2 64
• Environmental Testing 2.0 1.0 90
• Unit Profit Contribution $75 $110 —

In addition, a corporate supply agreement mandates that the facility must deliver at least 20 total controllers per week to keep its prime vendor certification, and customer demand for the Industrial model is capped at 40 units per week. Formulate the complete linear program.

Solution :

Define Decision Variables :

x 1 = number of Standard controllers assembled per week [units/week] x_1 = \text{number of Standard controllers assembled per week [units/week]} x 1 ​ = number of Standard controllers assembled per week [units/week]

x 2 = number of Industrial controllers assembled per week [units/week] x_2 = \text{number of Industrial controllers assembled per week [units/week]} x 2 ​ = number of Industrial controllers assembled per week [units/week]

Formulate the Objective Function :

Maximize total weekly profit contribution: Maximize Z = 75 x 1 + 110 x 2 \text{Maximize } Z = 75 x_1 + 110 x_2 Maximize Z = 75 x 1 ​ + 110 x 2 ​

Formulate Structural Constraints :

• SMT Placement capacity limit: 1.5 x 1 + 2.5 x 2 ≤ 120 1.5 x_1 + 2.5 x_2 \le 120 1.5 x 1 ​ + 2.5 x 2 ​ ≤ 120
• AOI Inspection capacity limit: 0.8 x 1 + 1.2 x 2 ≤ 64 0.8 x_1 + 1.2 x_2 \le 64 0.8 x 1 ​ + 1.2 x 2 ​ ≤ 64

Environmental Stress Testing capacity limit: 2.0 x 1 + 1.0 x 2 ≤ 90 2.0 x_1 + 1.0 x_2 \le 90 2.0 x 1 ​ + 1.0 x 2 ​ ≤ 90

• Minimum total production commitment: x 1 + x 2 ≥ 20 x_1 + x_2 \ge 20 x 1 ​ + x 2 ​ ≥ 20
• Maximum demand ceiling for Industrial model: x 2 ≤ 40 x_2 \le 40 x 2 ​ ≤ 40

• State Non-Negativity Constraints : x 1 ≥ 0 , x 2 ≥ 0 x_1 \ge 0, \quad x_2 \ge 0 x 1 ​ ≥ 0 , x 2 ​ ≥ 0
• Worked Example 8.1.2: Specialty Chemical Blending with Linearized Octane Specification

Problem : A refinery produces high-grade aviation fuel by blending two intermediate petroleum feedstocks: Alkylate ( x A x_A x A ​ ) and Reformate ( x R x_R x R ​ ). The chemical properties and unit costs per barrel are as follows:

• Feedstock Research Octane Number (RON) Vapor Pressure (psi) Cost per Barrel ($/bbl)
• Alkylate ( x A x_A x A ​ ) 98 4.5 $92
• Reformate ( x R x_R x R ​ ) 92 2.0 $80

The final blend must satisfy three quality standards:

Total production must be exactly 10,000 barrels.

The blended Research Octane Number (RON) must be at least 95.

The blended vapor pressure must not exceed 3.5 psi.

Formulate the linear programming model to minimize total blending cost.

Solution :

Decision Variables :

x A = barrels of Alkylate blended [bbl] x_A = \text{barrels of Alkylate blended [bbl]} x A ​ = barrels of Alkylate blended [bbl]

x R = barrels of Reformate blended [bbl] x_R = \text{barrels of Reformate blended [bbl]} x R ​ = barrels of Reformate blended [bbl]

Objective Function :

Minimize total cost: Minimize Z = 92 x A + 80 x R \text{Minimize } Z = 92 x_A + 80 x_R Minimize Z = 92 x A ​ + 80 x R ​

Structural Constraints :

Total volume requirement: x A + x R = 10,000 x_A + x_R = 10{,}000 x A ​ + x R ​ = 10 , 000

Minimum Octane Specification (Linearization): 98 x A + 92 x R x A + x R ≥ 95 \frac{98 x_A + 92 x_R}{x_A + x_R} \ge 95 x A ​ + x R ​ 98 x A ​ + 92 x R ​ ​ ≥ 95 Multiply across by ( x A + x R ) (x_A + x_R) ( x A ​ + x R ​ ) (valid since x A + x R = 10,000 > 0 x_A + x_R = 10{,}000 > 0 x A ​ + x R ​ = 10 , 000 > 0 ): 98 x A + 92 x R ≥ 95 ( x A + x R ) 98 x_A + 92 x_R \ge 95(x_A + x_R) 98 x A ​ + 92 x R ​ ≥ 95 ( x A ​ + x R ​ ) 98 x A + 92 x R ≥ 95 x A + 95 x R 98 x_A + 92 x_R \ge 95 x_A + 95 x_R 98 x A ​ + 92 x R ​ ≥ 95 x A ​ + 95 x R ​ Collect terms on the left: ( 98 − 95 ) x A + ( 92 − 95 ) x R ≥ 0 ⟹ 3 x A − 3 x R ≥ 0 (98 - 95) x_A + (92 - 95) x_R \ge 0 \implies 3 x_A - 3 x_R \ge 0 ( 98 − 95 ) x A ​ + ( 92 − 95 ) x R ​ ≥ 0 ⟹ 3 x A ​ − 3 x R ​ ≥ 0

Maximum Vapor Pressure Specification (Linearization): 4.5 x A + 2.0 x R x A + x R ≤ 3.5 \frac{4.5 x_A + 2.0 x_R}{x_A + x_R} \le 3.5 x A ​ + x R ​ 4.5 x A ​ + 2.0 x R ​ ​ ≤ 3.5 4.5 x A + 2.0 x R ≤ 3.5 ( x A + x R ) ⟹ ( 4.5 − 3.5 ) x A + ( 2.0 − 3.5 ) x R ≤ 0 4.5 x_A + 2.0 x_R \le 3.5(x_A + x_R) \implies (4.5 - 3.5) x_A + (2.0 - 3.5) x_R \le 0 4.5 x A ​ + 2.0 x R ​ ≤ 3.5 ( x A ​ + x R ​ ) ⟹ ( 4.5 − 3.5 ) x A ​ + ( 2.0 − 3.5 ) x R ​ ≤ 0 1.0 x A − 1.5 x R ≤ 0 1.0 x_A - 1.5 x_R \le 0 1.0 x A ​ − 1.5 x R ​ ≤ 0

• Non-Negativity : x A ≥ 0 , x R ≥ 0 x_A \ge 0, \quad x_R \ge 0 x A ​ ≥ 0 , x R ​ ≥ 0
• 6. NCEES Reference Handbook Tips & Realistic Exam Traps

The Unit Conversion Trap (Minutes vs. Hours) : One of the most frequent errors on the FE exam occurs when task times are given in minutes (e.g., 15 min/unit 15\text{ min/unit} 15 min/unit ) while available work center capacity is given in hours (e.g., 40 hours/week 40\text{ hours/week} 40 hours/week ). You must convert either minutes to hours ( 15 / 60 = 0.25 hr/unit 15/60 = 0.25\text{ hr/unit} 15/60 = 0.25 hr/unit ) or hours to minutes ( 40 × 60 = 2400 min 40 \times 60 = 2400\text{ min} 40 × 60 = 2400 min ). Mixing 15 x 1 + 25 x 2 ≤ 40 15 x_1 + 25 x_2 \le 40 15 x 1 ​ + 25 x 2 ​ ≤ 40 creates an immediate factor-of-60 error.

The Fractional Constraint Trap : Never submit an optimization model with variables in the denominator. A constraint like x 1 x 1 + x 2 ≥ 0.40 \frac{x_1}{x_1 + x_2} \ge 0.40 x 1 ​ + x 2 ​ x 1 ​ ​ ≥ 0.40 is technically non-linear in that format. Always cross-multiply to form 0.6 x 1 − 0.4 x 2 ≥ 0 0.6 x_1 - 0.4 x_2 \ge 0 0.6 x 1 ​ − 0.4 x 2 ​ ≥ 0 .

The Direction of Optimization in Cost vs. Profit : Double check whether the prompt specifies maximizing profit or minimizing cost. Selecting the wrong optimization sense will completely invert your solution.

Sign Restrictions for Net Balances : In cash-flow or temperature deviation models where variables can drop below zero, do not automatically append x j ≥ 0 x_j \ge 0 x j ​ ≥ 0 . Note whether a variable is unrestricted in sign (URS).