FE Mechanical

4. Engineering economics

Our FE Mechanical course is currently in development and is a work-in-progress.

Break-even analysis, risk analysis & accounting principles

7 min read

Break-even analysis

Break-even analysis determines the conditions under which two alternatives are equivalent, or when revenues equal costs.

Break-even point

The break-even quantity $Q$ is found by setting total revenue equal to total cost:

$T o t a l R e v e n u e = T o t a l C os t$

$pQ = C_{f} + c_{v} Q$

$Q = \frac{C _{f}}{p - c _{v}}$

where:

$p$ = selling price per unit
$c_{v}$ = variable cost per unit
$C_{f}$ = fixed costs

Comparing alternatives

When comparing two alternatives with different cost structures, the break-even point is where their total costs are equal:

$C_{f 1} + c_{v 1} Q = C_{f 2} + c_{v 2} Q$

$Q = \frac{C _{f 2} - C _{f 1}}{c _{v 1} - c _{v 2}}$

Benefit-cost analysis for public projects

Benefit-cost analysis is commonly used for public projects where benefits accrue to society rather than generating direct revenue.

Benefit-cost ratio:

$B / C = \frac{P W _{b e n e f i t s}}{P W _{cos t s}}$

Decision rule: Accept if B/C ≥ 1.0

Some analysts prefer to subtract operating costs from benefits to obtain a modified benefit-cost ratio:

$B / C_{m o d i f i e d} = \frac{P W _{b e n e f i t s} - P W _{O & M}}{P W _{ini t ia l cos t}}$

where O&M represents operating and maintenance costs.

When comparing mutually exclusive alternatives, use incremental analysis:

Rank alternatives by increasing cost
Compare each alternative to the previous acceptable one
Accept the higher-cost alternative if the incremental B/C ratio exceeds 1.0

Public project analysis faces unique challenges:

Quantifying benefits: Many public benefits (safety, environmental quality, quality of life) are difficult to monetize
Discount rate selection: The appropriate discount rate for public projects is debated
Distribution of benefits: Benefits and costs may fall on different groups
Intangible factors: Some important considerations cannot be quantified

Sensitivity, risk, and uncertainty analysis

Sensitivity analysis examines how changes in input variables affect the outcome. It answers questions like: “What if the interest rate is 12% instead of 10%?” or “What if demand is 20% lower than expected?”

Procedure:

Identify key input variables
Vary each variable over a reasonable range
Calculate the effect on the decision criterion
Identify which variables have the greatest impact

Spider diagrams graphically display sensitivity analysis results, showing how the decision criterion changes as each input variable is varied from its base case value.

Tornado diagrams rank variables by their impact on the outcome, with the most sensitive variables at the top.

Risk analysis

Risk analysis incorporates probability distributions for uncertain variables. Monte Carlo simulation is a common technique:

Define probability distributions for uncertain inputs
Randomly sample from each distribution
Calculate the outcome for each sample
Repeat many times to generate a distribution of outcomes
Analyze the results (mean, standard deviation, probability of loss, etc.)

The expected value is the probability-weighted average of possible outcomes:

$E [X] = i \sum p_{i} \times X_{i}$

where $p_{i}$ is the probability of outcome $X_{i}$ .

Uncertainty analysis

When probabilities cannot be assigned, uncertainty analysis uses decision criteria such as:

Maximax: Choose the alternative with the best possible outcome (optimistic)
Maximin: Choose the alternative with the best worst-case outcome (pessimistic)
Laplace: Assume all outcomes are equally likely
Minimax regret: Minimize the maximum opportunity cost

Decision trees

Decision trees provide a structured approach to analyzing decisions under uncertainty:

Identify decision points (squares) and chance events (circles)
Assign probabilities to chance outcomes
Assign values to final outcomes
Calculate expected values by working backward through the tree
Select the alternative with the highest expected value

Accounting principles

Understanding basic accounting principles helps engineers communicate with financial professionals and interpret financial data.

The accounting equation

$A sse t s = L iabi l i t i es + Eq u i t y$

This fundamental equation must always balance.

Double-entry bookkeeping

Every transaction affects at least two accounts, maintaining the balance of the accounting equation. Transactions are recorded as:

Debits: Increases in assets or decreases in liabilities/equity
Credits: Decreases in assets or increases in liabilities/equity

Cash vs. accrual accounting

Cash basis: Records transactions when cash changes hands
Accrual basis: Records transactions when they are earned or incurred, regardless of cash flow

Most businesses use accrual accounting for financial reporting.

Financial statements

Definitions

Balance sheet: Shows assets, liabilities, and equity at a point in time. Provides a snapshot of the company’s financial position.
Income statement: Shows revenues, expenses, and profit over a period. Also called the profit and loss statement (P&L).
Cash flow statement: Shows sources and uses of cash over a period. Categorizes cash flows as operating, investing, or financing activities.

Cost accounting

Cost accounting provides detailed cost information for internal decision-making.

Cost classifications

Direct costs: Traceable to a specific product or project:

Direct materials
Direct labor

Indirect costs (overhead): Not directly traceable:

Utilities
Supervision
Depreciation
Rent
Insurance

Fixed costs: Do not vary with production volume:

Rent
Insurance
Salaries
Depreciation

**Variable costs:**Vary proportionally with production:

Raw materials
Direct labor (sometimes)
Utilities (partially)
Shipping

Semi-variable costs: Have both fixed and variable components:

Utilities (base charge plus usage)
Maintenance (scheduled plus demand)

Standard costs

Standard costs are predetermined costs used for planning and control:

$Va r ian ce = A c t u a l C os t - St an d a r d C os t$

Favorable variances indicate costs below standard; unfavorable variances indicate costs above standard.

Activity-based costing (ABC)

ABC allocates overhead costs based on activities that drive costs, rather than simple volume measures. This provides more accurate product costs when overhead is significant and products differ in complexity.

Steps in ABC:

Identify activities that consume resources
Assign costs to activity cost pools
Identify cost drivers for each activity
Calculate activity rates
Assign costs to products based on activity consumption

Summary

Engineering economics provides the essential framework for making sound financial decisions in engineering contexts. The key concepts covered in this lecture include:

Time value of money: The fundamental principle that money has different values at different times due to its earning potential
Cash flow analysis: Systematic representation of financial transactions over time using standard patterns and diagrams
Equivalence calculations: Converting cash flows between different time periods using discount factors to enable meaningful comparisons
Depreciation methods: Allocating asset costs over useful life for tax and accounting purposes using straight-line, SOYD, declining balance, or MACRS methods
Alternative comparison: Methods for selecting among competing investment options, including present worth, annual worth, rate of return, and benefit-cost ratio
Inflation adjustment: Accounting for changing purchasing power over time using real and nominal interest rates
Risk and sensitivity analysis: Dealing with uncertainty in economic decisions through sensitivity analysis, Monte Carlo simulation, and decision trees

These principles enable engineers to:

Justify capital investments with rigorous financial analysis
Compare alternatives with different cost and benefit patterns
Communicate effectively with financial professionals
Make decisions that optimize both technical and economic performance

The ability to perform engineering economic analysis is essential for engineers in all disciplines. Technical excellence alone is insufficient; projects must also be economically viable. By mastering these concepts, engineers can ensure that their technical solutions create genuine value for their organizations and society.

As you apply these principles, remember that engineering economic analysis involves estimates and assumptions about the future. The results are only as good as the inputs. Sensitivity analysis helps identify which assumptions are most critical, and good judgment remains essential in interpreting results and making final decisions.

Next | 5. Electricity & magnetism

Break-even analysis, risk analysis & accounting principles

Break-even analysis

Break-even analysis determines the conditions under which two alternatives are equivalent, or when revenues equal costs.

Break-even point

The break-even quantity $Q$ is found by setting total revenue equal to total cost:

$T o t a l R e v e n u e = T o t a l C os t$

$pQ = C_{f} + c_{v} Q$

$Q = \frac{C _{f}}{p - c _{v}}$

where:

$p$ = selling price per unit
$c_{v}$ = variable cost per unit
$C_{f}$ = fixed costs

Comparing alternatives

When comparing two alternatives with different cost structures, the break-even point is where their total costs are equal:

$C_{f 1} + c_{v 1} Q = C_{f 2} + c_{v 2} Q$

$Q = \frac{C _{f 2} - C _{f 1}}{c _{v 1} - c _{v 2}}$

Benefit-cost analysis for public projects

Benefit-cost analysis is commonly used for public projects where benefits accrue to society rather than generating direct revenue.

Benefit-cost ratio:

$B / C = \frac{P W _{b e n e f i t s}}{P W _{cos t s}}$

Decision rule: Accept if B/C ≥ 1.0

Some analysts prefer to subtract operating costs from benefits to obtain a modified benefit-cost ratio:

$B / C_{m o d i f i e d} = \frac{P W _{b e n e f i t s} - P W _{O & M}}{P W _{ini t ia l cos t}}$

where O&M represents operating and maintenance costs.

When comparing mutually exclusive alternatives, use incremental analysis:

Rank alternatives by increasing cost
Compare each alternative to the previous acceptable one
Accept the higher-cost alternative if the incremental B/C ratio exceeds 1.0

Public project analysis faces unique challenges:

Quantifying benefits: Many public benefits (safety, environmental quality, quality of life) are difficult to monetize
Discount rate selection: The appropriate discount rate for public projects is debated
Distribution of benefits: Benefits and costs may fall on different groups
Intangible factors: Some important considerations cannot be quantified

Sensitivity, risk, and uncertainty analysis

Procedure:

Identify key input variables
Vary each variable over a reasonable range
Calculate the effect on the decision criterion
Identify which variables have the greatest impact

Spider diagrams graphically display sensitivity analysis results, showing how the decision criterion changes as each input variable is varied from its base case value.

Tornado diagrams rank variables by their impact on the outcome, with the most sensitive variables at the top.

Risk analysis

Risk analysis incorporates probability distributions for uncertain variables. Monte Carlo simulation is a common technique:

Define probability distributions for uncertain inputs
Randomly sample from each distribution
Calculate the outcome for each sample
Repeat many times to generate a distribution of outcomes
Analyze the results (mean, standard deviation, probability of loss, etc.)

The expected value is the probability-weighted average of possible outcomes:

$E [X] = i \sum p_{i} \times X_{i}$

where $p_{i}$ is the probability of outcome $X_{i}$ .

Uncertainty analysis

When probabilities cannot be assigned, uncertainty analysis uses decision criteria such as:

Maximax: Choose the alternative with the best possible outcome (optimistic)
Maximin: Choose the alternative with the best worst-case outcome (pessimistic)
Laplace: Assume all outcomes are equally likely
Minimax regret: Minimize the maximum opportunity cost

Decision trees

Decision trees provide a structured approach to analyzing decisions under uncertainty:

Identify decision points (squares) and chance events (circles)
Assign probabilities to chance outcomes
Assign values to final outcomes
Calculate expected values by working backward through the tree
Select the alternative with the highest expected value

Accounting principles

Understanding basic accounting principles helps engineers communicate with financial professionals and interpret financial data.

The accounting equation

$A sse t s = L iabi l i t i es + Eq u i t y$

This fundamental equation must always balance.

Double-entry bookkeeping

Every transaction affects at least two accounts, maintaining the balance of the accounting equation. Transactions are recorded as:

Debits: Increases in assets or decreases in liabilities/equity
Credits: Decreases in assets or increases in liabilities/equity

Cash vs. accrual accounting

Cash basis: Records transactions when cash changes hands
Accrual basis: Records transactions when they are earned or incurred, regardless of cash flow

Most businesses use accrual accounting for financial reporting.

Financial statements

Definitions

Balance sheet: Shows assets, liabilities, and equity at a point in time. Provides a snapshot of the company’s financial position.
Income statement: Shows revenues, expenses, and profit over a period. Also called the profit and loss statement (P&L).
Cash flow statement: Shows sources and uses of cash over a period. Categorizes cash flows as operating, investing, or financing activities.

Cost accounting

Cost accounting provides detailed cost information for internal decision-making.

Cost classifications

Direct costs: Traceable to a specific product or project:

Direct materials
Direct labor

Indirect costs (overhead): Not directly traceable:

Utilities
Supervision
Depreciation
Rent
Insurance

Fixed costs: Do not vary with production volume:

Rent
Insurance
Salaries
Depreciation

**Variable costs:**Vary proportionally with production:

Raw materials
Direct labor (sometimes)
Utilities (partially)
Shipping

Semi-variable costs: Have both fixed and variable components:

Utilities (base charge plus usage)
Maintenance (scheduled plus demand)

Standard costs

Standard costs are predetermined costs used for planning and control:

$Va r ian ce = A c t u a l C os t - St an d a r d C os t$

Favorable variances indicate costs below standard; unfavorable variances indicate costs above standard.

Activity-based costing (ABC)

Steps in ABC:

Identify activities that consume resources
Assign costs to activity cost pools
Identify cost drivers for each activity
Calculate activity rates
Assign costs to products based on activity consumption

Summary

Engineering economics provides the essential framework for making sound financial decisions in engineering contexts. The key concepts covered in this lecture include:

Time value of money: The fundamental principle that money has different values at different times due to its earning potential
Cash flow analysis: Systematic representation of financial transactions over time using standard patterns and diagrams
Equivalence calculations: Converting cash flows between different time periods using discount factors to enable meaningful comparisons
Depreciation methods: Allocating asset costs over useful life for tax and accounting purposes using straight-line, SOYD, declining balance, or MACRS methods
Alternative comparison: Methods for selecting among competing investment options, including present worth, annual worth, rate of return, and benefit-cost ratio
Inflation adjustment: Accounting for changing purchasing power over time using real and nominal interest rates
Risk and sensitivity analysis: Dealing with uncertainty in economic decisions through sensitivity analysis, Monte Carlo simulation, and decision trees

These principles enable engineers to:

Justify capital investments with rigorous financial analysis
Compare alternatives with different cost and benefit patterns
Communicate effectively with financial professionals
Make decisions that optimize both technical and economic performance

Break-even analysis

Break-even analysis determines the conditions under which two alternatives are equivalent, or when revenues equal costs.

Break-even point

The break-even quantity $Q$ is found by setting total revenue equal to total cost:

$T o t a l R e v e n u e = T o t a l C os t$

$pQ = C_{f} + c_{v} Q$

$Q = \frac{C _{f}}{p - c _{v}}$

where:

$p$ = selling price per unit
$c_{v}$ = variable cost per unit
$C_{f}$ = fixed costs

Comparing alternatives

When comparing two alternatives with different cost structures, the break-even point is where their total costs are equal:

$C_{f 1} + c_{v 1} Q = C_{f 2} + c_{v 2} Q$

$Q = \frac{C _{f 2} - C _{f 1}}{c _{v 1} - c _{v 2}}$

Benefit-cost analysis for public projects

Benefit-cost analysis is commonly used for public projects where benefits accrue to society rather than generating direct revenue.

Benefit-cost ratio:

$B / C = \frac{P W _{b e n e f i t s}}{P W _{cos t s}}$

Decision rule: Accept if B/C ≥ 1.0

Some analysts prefer to subtract operating costs from benefits to obtain a modified benefit-cost ratio:

$B / C_{m o d i f i e d} = \frac{P W _{b e n e f i t s} - P W _{O & M}}{P W _{ini t ia l cos t}}$

where O&M represents operating and maintenance costs.

When comparing mutually exclusive alternatives, use incremental analysis:

Rank alternatives by increasing cost
Compare each alternative to the previous acceptable one
Accept the higher-cost alternative if the incremental B/C ratio exceeds 1.0

Public project analysis faces unique challenges:

Quantifying benefits: Many public benefits (safety, environmental quality, quality of life) are difficult to monetize
Discount rate selection: The appropriate discount rate for public projects is debated
Distribution of benefits: Benefits and costs may fall on different groups
Intangible factors: Some important considerations cannot be quantified

Sensitivity, risk, and uncertainty analysis

Procedure:

Identify key input variables
Vary each variable over a reasonable range
Calculate the effect on the decision criterion
Identify which variables have the greatest impact

Spider diagrams graphically display sensitivity analysis results, showing how the decision criterion changes as each input variable is varied from its base case value.

Tornado diagrams rank variables by their impact on the outcome, with the most sensitive variables at the top.

Risk analysis

Risk analysis incorporates probability distributions for uncertain variables. Monte Carlo simulation is a common technique:

Define probability distributions for uncertain inputs
Randomly sample from each distribution
Calculate the outcome for each sample
Repeat many times to generate a distribution of outcomes
Analyze the results (mean, standard deviation, probability of loss, etc.)

The expected value is the probability-weighted average of possible outcomes:

$E [X] = i \sum p_{i} \times X_{i}$

where $p_{i}$ is the probability of outcome $X_{i}$ .

Uncertainty analysis

When probabilities cannot be assigned, uncertainty analysis uses decision criteria such as:

Maximax: Choose the alternative with the best possible outcome (optimistic)
Maximin: Choose the alternative with the best worst-case outcome (pessimistic)
Laplace: Assume all outcomes are equally likely
Minimax regret: Minimize the maximum opportunity cost

Decision trees

Decision trees provide a structured approach to analyzing decisions under uncertainty:

Identify decision points (squares) and chance events (circles)
Assign probabilities to chance outcomes
Assign values to final outcomes
Calculate expected values by working backward through the tree
Select the alternative with the highest expected value

Accounting principles

Understanding basic accounting principles helps engineers communicate with financial professionals and interpret financial data.

The accounting equation

$A sse t s = L iabi l i t i es + Eq u i t y$

This fundamental equation must always balance.

Double-entry bookkeeping

Every transaction affects at least two accounts, maintaining the balance of the accounting equation. Transactions are recorded as:

Debits: Increases in assets or decreases in liabilities/equity
Credits: Decreases in assets or increases in liabilities/equity

Cash vs. accrual accounting

Cash basis: Records transactions when cash changes hands
Accrual basis: Records transactions when they are earned or incurred, regardless of cash flow

Most businesses use accrual accounting for financial reporting.

Financial statements

Definitions

Balance sheet: Shows assets, liabilities, and equity at a point in time. Provides a snapshot of the company’s financial position.
Income statement: Shows revenues, expenses, and profit over a period. Also called the profit and loss statement (P&L).
Cash flow statement: Shows sources and uses of cash over a period. Categorizes cash flows as operating, investing, or financing activities.

Cost accounting

Cost accounting provides detailed cost information for internal decision-making.

Cost classifications

Direct costs: Traceable to a specific product or project:

Direct materials
Direct labor

Indirect costs (overhead): Not directly traceable:

Utilities
Supervision
Depreciation
Rent
Insurance

Fixed costs: Do not vary with production volume:

Rent
Insurance
Salaries
Depreciation

**Variable costs:**Vary proportionally with production:

Raw materials
Direct labor (sometimes)
Utilities (partially)
Shipping

Semi-variable costs: Have both fixed and variable components:

Utilities (base charge plus usage)
Maintenance (scheduled plus demand)

Standard costs

Standard costs are predetermined costs used for planning and control:

$Va r ian ce = A c t u a l C os t - St an d a r d C os t$

Favorable variances indicate costs below standard; unfavorable variances indicate costs above standard.

Activity-based costing (ABC)

Steps in ABC:

Identify activities that consume resources
Assign costs to activity cost pools
Identify cost drivers for each activity
Calculate activity rates
Assign costs to products based on activity consumption

Summary

Engineering economics provides the essential framework for making sound financial decisions in engineering contexts. The key concepts covered in this lecture include:

Time value of money: The fundamental principle that money has different values at different times due to its earning potential
Cash flow analysis: Systematic representation of financial transactions over time using standard patterns and diagrams
Equivalence calculations: Converting cash flows between different time periods using discount factors to enable meaningful comparisons
Depreciation methods: Allocating asset costs over useful life for tax and accounting purposes using straight-line, SOYD, declining balance, or MACRS methods
Alternative comparison: Methods for selecting among competing investment options, including present worth, annual worth, rate of return, and benefit-cost ratio
Inflation adjustment: Accounting for changing purchasing power over time using real and nominal interest rates
Risk and sensitivity analysis: Dealing with uncertainty in economic decisions through sensitivity analysis, Monte Carlo simulation, and decision trees

These principles enable engineers to:

Justify capital investments with rigorous financial analysis
Compare alternatives with different cost and benefit patterns
Communicate effectively with financial professionals
Make decisions that optimize both technical and economic performance